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?

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).
