# Bounded sequence of norms for operators between Banach spaces

Looking at this problem:

Let $$\lbrace T_{n} \rbrace_{n=1}^{\infty}$$ be a sequence in $$\mathcal{L}(X,Y)$$, where $$X$$ and $$Y$$ are Banach spaces. Prove that the sequence $$\lbrace ||T_{n}|| \rbrace_{n=1}^{\infty}$$ is bounded if and only if for each $$x \in X$$, the sequence $$\lbrace T_{n}(x) \rbrace_{n=1}^{\infty}$$ is bounded in Y.

I'm not exactly sure how to even start it. I suspect that I need to use uniform boundedness, but I cant see exactly how it fits. Any help is appreciated!

Suppose that for every $$x$$ the sequence $$T_n(x)$$ is bounded, the uniform boundedness theorem implies that the sequence $$\|T_n\|$$ is bounded.
On the other hand, suppose that the sequence $$\|T_n\|$$ is bounded, there exists $$C$$ such that for every $$n$$, $$\|T_n\|, we deduce that $$\|T_n(x)\|\leq C\|x\|$$ for every $$n$$.