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Proof FoundationsQuantifier NegationArchimedean LanguageLogical Equivalence

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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Proof FoundationsContrapositiveTruth ValuesImplication

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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Proof FoundationsRational ClosureIrrational NumbersAdditive Inverses

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsParityDivisibilityContrapositive

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsSqrt TwoLowest TermsParity Descent

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsSet ExtensionalityDE Morgan LawComplements

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsPower SetSubset RelationSingleton Witness

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:008Preimage of indexed intersections

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Proof FoundationsPreimageIndexed IntersectionFunction Membership

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Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

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RA:009Image of an intersection

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Proof FoundationsSet ImageInjective FunctionsIntersection Counterexample

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Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

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RA:010Injective compositions

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Proof FoundationsFunction CompositionInjectivityRange Restriction

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:011Surjective compositions

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Proof FoundationsSurjective FunctionsComposition RangeFinite Counterexample

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:012Congruence modulo five

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Proof FoundationsEquivalence RelationModular CongruenceInteger Divisibility

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:013Equality of equivalence classes

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Proof FoundationsEquivalence ClassesSet ExtensionalityTransitive Relations

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:014Sum of the first odd integers

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Proof FoundationsMathematical InductionFinite SumsOdd Numbers

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:015Exponential versus linear growth

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Proof FoundationsInduction InequalityExponential GrowthEquality Cases

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:016Existence of a prime divisor

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Proof FoundationsStrong InductionPrime DivisorsFactorization

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:017A recursive affine sequence

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Proof FoundationsRecursive SequenceClosed FormInduction

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:018Finite geometric sum

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Proof FoundationsGeometric SeriesFinite SumInduction Algebra

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:019Enumerating the integers

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Proof FoundationsCountable SetsInteger EnumerationSurjective Map

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:020Diagonal enumeration of pairs

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Proof FoundationsCartesian ProductDiagonal EnumerationCountability

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:021Countable union theorem

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Proof FoundationsCountable UnionEnumerationsPairing Function

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:022Countability of the rationals

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Proof FoundationsRational NumbersQuotient RepresentationCountable Image

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:023Cantor diagonal binary sequence

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Proof FoundationsCantor DiagonalBinary SequencesUncountability

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:024Negation of sequence convergence

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Proof FoundationsSequence ConvergenceEpsilon NegationQuantifier Order

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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RA:025Quantifier exchange trap

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Proof FoundationsQuantifier OrderUnbounded Real LineLogical Counterexample

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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RA:026Difference of a union

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Proof FoundationsSet DifferenceDE Morgan LawMembership Equivalence

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:027Image after preimage

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Proof FoundationsImage PreimageFunction RangeSet Equality

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Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

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RA:028A unique affine root

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Proof FoundationsExistence UniquenessAffine EquationNonzero Coefficient

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:029A cubic divisible by six

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Proof FoundationsInteger DivisibilityConsecutive IntegersModular Arithmetic

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:030Pointwise is not uniform

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Proof FoundationsWitness DependenceArchimedean ChoiceUniform Quantifier

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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RA:031Triangle inequality from cases

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Foundations And CompletenessAbsolute ValueTriangle InequalitySign Cases

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:032Reverse triangle inequality

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Foundations And CompletenessReverse TriangleAbsolute ValueDistance Estimate

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:033Absolute-value neighborhoods

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Foundations And CompletenessAbsolute ValueOpen IntervalInequality Translation

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Ordered fields, bounds, and completenessReal Analysis I · Concept & counterexample

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RA:034Archimedean reciprocal estimate

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Foundations And CompletenessArchimedean PropertyReciprocal BoundEpsilon Estimate

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:035Integer part of a real number

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Foundations And CompletenessFloor PropertyWell OrderingArchimedean Property

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:036Density of the rationals

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Foundations And CompletenessRational DensityArchimedean PropertyFloor Construction

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:037Density of irrational numbers

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Foundations And CompletenessIrrational DensityRational DensityScaling Translation

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:038Supremum of a half-open interval

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Foundations And CompletenessSupremumUpper BoundHalf Open Interval

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:039Epsilon characterization of supremum

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Foundations And CompletenessSupremum CharacterizationEpsilon WitnessLeast Upper Bound

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:040Uniqueness of the supremum

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Foundations And CompletenessSupremum UniquenessAntisymmetryUpper Bounds

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:041Supremum of a Minkowski sum

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Foundations And CompletenessMinkowski SumSupremum AlgebraEpsilon Splitting

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:042Positive scaling of a supremum

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Foundations And CompletenessScaled SetSupremum AlgebraPositive Scalar

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:043Infimum through negation

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Foundations And CompletenessInfimumReflected SetOrder Reversal

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:044Supremum of a finite union

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Foundations And CompletenessFinite UnionSupremumMaximum Upper Bound

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:045Nested interval theorem

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Foundations And CompletenessNested IntervalsSupremumCompleteness

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:046Existence of square roots

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Foundations And CompletenessSquare Root ExistenceSupremum ConstructionAlgebraic Perturbation

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:047A supremum not attained

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Foundations And CompletenessNonattained SupremumSquare RootStrict Inequality

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:048The rational square-root gap

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Foundations And CompletenessRational IncompletenessSqrt TwoSupremum Gap

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:049Monotone convergence theorem

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Foundations And CompletenessMonotone SequenceBounded SequenceSupremum Limit

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:050A bounded geometric partial sum

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Foundations And CompletenessGeometric Partial SumsMonotone ConvergenceEpsilon Tail

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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Problem library

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Proof FoundationsQuantifier NegationArchimedean LanguageLogical Equivalence

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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Proof FoundationsContrapositiveTruth ValuesImplication

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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Proof FoundationsRational ClosureIrrational NumbersAdditive Inverses

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsParityDivisibilityContrapositive

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsSqrt TwoLowest TermsParity Descent

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsSet ExtensionalityDE Morgan LawComplements

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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Proof FoundationsPower SetSubset RelationSingleton Witness

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:008Preimage of indexed intersections

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Proof FoundationsPreimageIndexed IntersectionFunction Membership

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Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

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RA:009Image of an intersection

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Proof FoundationsSet ImageInjective FunctionsIntersection Counterexample

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Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

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RA:010Injective compositions

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Proof FoundationsFunction CompositionInjectivityRange Restriction

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:011Surjective compositions

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Proof FoundationsSurjective FunctionsComposition RangeFinite Counterexample

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:012Congruence modulo five

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Proof FoundationsEquivalence RelationModular CongruenceInteger Divisibility

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:013Equality of equivalence classes

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Proof FoundationsEquivalence ClassesSet ExtensionalityTransitive Relations

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:014Sum of the first odd integers

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Proof FoundationsMathematical InductionFinite SumsOdd Numbers

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:015Exponential versus linear growth

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Proof FoundationsInduction InequalityExponential GrowthEquality Cases

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:016Existence of a prime divisor

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Proof FoundationsStrong InductionPrime DivisorsFactorization

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:017A recursive affine sequence

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Proof FoundationsRecursive SequenceClosed FormInduction

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:018Finite geometric sum

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Proof FoundationsGeometric SeriesFinite SumInduction Algebra

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:019Enumerating the integers

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Proof FoundationsCountable SetsInteger EnumerationSurjective Map

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:020Diagonal enumeration of pairs

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Proof FoundationsCartesian ProductDiagonal EnumerationCountability

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:021Countable union theorem

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Proof FoundationsCountable UnionEnumerationsPairing Function

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:022Countability of the rationals

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Proof FoundationsRational NumbersQuotient RepresentationCountable Image

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:023Cantor diagonal binary sequence

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Proof FoundationsCantor DiagonalBinary SequencesUncountability

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:024Negation of sequence convergence

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Proof FoundationsSequence ConvergenceEpsilon NegationQuantifier Order

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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RA:025Quantifier exchange trap

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Proof FoundationsQuantifier OrderUnbounded Real LineLogical Counterexample

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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RA:026Difference of a union

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Proof FoundationsSet DifferenceDE Morgan LawMembership Equivalence

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:027Image after preimage

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Proof FoundationsImage PreimageFunction RangeSet Equality

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Proof language, sets, functions, and quantifiersReal Analysis I · Calculation

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RA:028A unique affine root

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Proof FoundationsExistence UniquenessAffine EquationNonzero Coefficient

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:029A cubic divisible by six

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Proof FoundationsInteger DivisibilityConsecutive IntegersModular Arithmetic

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Proof language, sets, functions, and quantifiersReal Analysis I · Proof

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RA:030Pointwise is not uniform

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Proof FoundationsWitness DependenceArchimedean ChoiceUniform Quantifier

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Proof language, sets, functions, and quantifiersReal Analysis I · Concept & counterexample

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RA:031Triangle inequality from cases

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Foundations And CompletenessAbsolute ValueTriangle InequalitySign Cases

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:032Reverse triangle inequality

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Foundations And CompletenessReverse TriangleAbsolute ValueDistance Estimate

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:033Absolute-value neighborhoods

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Foundations And CompletenessAbsolute ValueOpen IntervalInequality Translation

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Ordered fields, bounds, and completenessReal Analysis I · Concept & counterexample

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RA:034Archimedean reciprocal estimate

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Foundations And CompletenessArchimedean PropertyReciprocal BoundEpsilon Estimate

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:035Integer part of a real number

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Foundations And CompletenessFloor PropertyWell OrderingArchimedean Property

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:036Density of the rationals

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Foundations And CompletenessRational DensityArchimedean PropertyFloor Construction

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:037Density of irrational numbers

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Foundations And CompletenessIrrational DensityRational DensityScaling Translation

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:038Supremum of a half-open interval

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Foundations And CompletenessSupremumUpper BoundHalf Open Interval

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:039Epsilon characterization of supremum

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Foundations And CompletenessSupremum CharacterizationEpsilon WitnessLeast Upper Bound

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:040Uniqueness of the supremum

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Foundations And CompletenessSupremum UniquenessAntisymmetryUpper Bounds

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:041Supremum of a Minkowski sum

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Foundations And CompletenessMinkowski SumSupremum AlgebraEpsilon Splitting

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:042Positive scaling of a supremum

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Foundations And CompletenessScaled SetSupremum AlgebraPositive Scalar

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:043Infimum through negation

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Foundations And CompletenessInfimumReflected SetOrder Reversal

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:044Supremum of a finite union

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Foundations And CompletenessFinite UnionSupremumMaximum Upper Bound

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:045Nested interval theorem

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Foundations And CompletenessNested IntervalsSupremumCompleteness

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:046Existence of square roots

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Foundations And CompletenessSquare Root ExistenceSupremum ConstructionAlgebraic Perturbation

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:047A supremum not attained

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Foundations And CompletenessNonattained SupremumSquare RootStrict Inequality

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Ordered fields, bounds, and completenessReal Analysis I · Calculation

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RA:048The rational square-root gap

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Foundations And CompletenessRational IncompletenessSqrt TwoSupremum Gap

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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RA:049Monotone convergence theorem

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Foundations And CompletenessMonotone SequenceBounded SequenceSupremum Limit

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Ordered fields, bounds, and completenessReal Analysis I · Theorem

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RA:050A bounded geometric partial sum

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Foundations And CompletenessGeometric Partial SumsMonotone ConvergenceEpsilon Tail

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Ordered fields, bounds, and completenessReal Analysis I · Proof

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Problem results

RA:001Negating alternating quantifiers

RA:002Contrapositive versus converse

RA:003Rational plus irrational

RA:004Even square forces even integer

RA:005Irrationality of square root two

RA:006De Morgan by element chasing

RA:007Power-set inclusion

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Problem results

RA:001Negating alternating quantifiers

RA:002Contrapositive versus converse

RA:003Rational plus irrational

RA:004Even square forces even integer

RA:005Irrationality of square root two

RA:006De Morgan by element chasing

RA:007Power-set inclusion