Question: Let (fk) be a sequence on [a, b] such that each fk is differentiable on [a, b] and f1/k(x) g(x) with |f1/k(x)| < K

Let (fk) be a sequence on [a, b] such that each fk is differentiable on [a, b] and f1/k(x) → g(x) with |f1/k(x)| < K for all x ∈ [a, b]. Show that the sequence (fk(x)) either converges for all x ∈ [a, b] or it diverges for all x ∈ [a, b].

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