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

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?
