Suppose that ⟨fn⟩ is a sequence of real valued differentiable functions on [0, 1], ⟨fn(0)⟩ is bounded, and that ⟨f′n⟩ is uniformly bounded on [0, 1]. Show that ⟨fn⟩ is uniformly bounded on [0, 1] and some subsequence of ⟨fn⟩ converges uniformly on [0, 1].
I didn't really have much problem starting off from uniform convergence to uniform boundedness, but the opposite seems a bit tricky... How shall we prove this?