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Prove that if {f_n:R→R} is a sequence of continuously differentiable functions such that the sequence of derivatives {f_n^':R→R} is uniformly convergent and the sequence {f_n (0)} is also convergent, then {f_n:R→R} is pointwise convergent. Is the assumption that the sequence {f_n (0)} converges necessary?

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A sequence of continuously differentiable functions is examined.

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Proof:
Since is continuous differentiable, then we have .
Now we fix some , , then for any , since is convergent, then we can find some ...

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