Real Analysis II · Uniform Convergence
Derivative interchange theorem
Problem
Let a<b and x₀∈[a,b]. Suppose f_n∈C¹[a,b], the scalar sequence f_n(x₀) converges, and f_n' converges uniformly on [a,b] to g. Prove f_n converges uniformly to a differentiable f with f'=g on (a,b).
Hint
Write every f_n relative to the common base point x₀ using the fundamental theorem.
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