Real Analysis I · Topology & Compactness
Why closedness alone is insufficient
Problem
Give nonempty nested closed subsets of $\mathbb R$ whose intersection is empty, and identify the missing compactness condition.
Hint
Take $F_n=[n,\infty)$.
Check your work
Work the problem yourself first. Then open it in Training to check your answer and read the full worked solution.
The answer check and full solution for this problem come with ProofAnvil Practice membership ($19 USD monthly). See membership. Or start with the free Real Analysis I sample problem: Try the free sample problem.
More Real Analysis I practice problems
- Negating alternating quantifiersProof Foundations
- Injective compositionsProof Foundations
- Integer part of a real numberFoundations & Completeness
- Shrinking nested intervalsFoundations & Completeness
- Compact sets have finite epsilon-netsTopology & Compactness
- Comparison using bounded partial sumsSequences & Series
- Summable increments force sequence convergenceSequences & Series
- Uniform limits preserve a common Lipschitz boundContinuity
- Connected preimages need not be connectedContinuity