I've been struggling with this for a bit and was wondering if anyone can give me a hint:
Suppose $\{ T_n\}_1^\infty\subset \mathcal{L}\{ X,Y\}$ is a sequence of bounded operators from a banach space $X$ to a normed space $Y$, such that $\forall x\in X:\lim_n T_n x=0$. Prove that $\lim_n(\sup\{\Vert T_n x\Vert:x\in K\})=0$ on any compact set $K\subset X$.
As far as I can see, the fact that $T_n$ attains it's maximum on $K$ for any $n$ is not enough to prove this one, since it is possible that there exists an $\epsilon>0$ such that for any $N$ $\exists n>N$ such that $\Vert T_nx\Vert>\epsilon$ for some $x\in K$, and diminishes to 0 later on. Plus, this kind of solution doesn't employ the fact the $X$ is complete.
I've been trying to work this question out as a varient of the Principle of Uniform Boundedness, but I keep getting stuck. Anyway I would be very thankful if anyone could tell me if I'm on the right track with this.
Cheers
P.S This is homework