If two sequences of functions $f_k$ and $g_k$ converges pointwise to the same limit $F$, and one of the convergence is uniform, is the other convergence necessarily uniform?
Edit: I guess I should throw in some conditions to be applicable to my situation. Let's assume that each of $f_k$ is continuous, and converges uniformly to $F$, thus makes $F$ continuous as well.
Let $f_{k}=0$ and $g_{k}(x)=x^{n}$ for $x\in[0,1)$, $g_{k}(1)=0$. $(g_{k})$ does not converge uniformly to $F=0$ on $[0,1]$.