Is that family of linear transformations $(T_{n})_{n\in \mathbb{N}}$ bounded?

Question

Let $X$ and $Y$ Banach spaces e over $\mathbb{K}$ and $T_{n}$ $B(X,Y)$, $n \in \mathbb{N}$ then

given $x \in X$ and $f \in B(Y,\mathbb{K})$, the sequence $f(T_{n}(x))$ is bounded in $\mathbb{K}$ then the sequence $(T_{n})$ is bounded in $B(X,Y)$.

any hint of how can i do this?