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If $f$ is continuous ,$F_n$ is pointwise convergent to $f$ and $F_n$ is uniformly convergent. Then $F_n$ in uniformly convergent to $f$.

To me, this result makes perfect sense. But I can not seem to find it in my analysis notes.

Is this implication correct?

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    Have you tried to prove the negation? Maybe you will immediately get the answer. – 2017-02-04

2 Answers 2

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If $F_n$ converges uniformly to a function $g$, then it is also pointwise convergent to $g$. Since $F_n$ is also pointwise convergent to $f$, we have $f=g$ (pointwise limit is unique).

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Your implication is correcto. Suppose that the sequence converges uniformly to a función $g$ and show that for all $\epsilon>0$, and $x$ in the domain, the difference between $f(x)$ and $g(x)$ is less than $\epsilon$. What can we conclude?