Sequence of linear maps between Banach Spaces and Banach-Steinhaus theorem.

Question

I have some problems proving the following :

Let $T,T_1,T_2,\dots$ linear maps between $E,F$, two Banach spaces (on the same field $\mathbb{R}$ or $\mathbb{C}$),$B \subset E$ s.t. $\overline{Span(B)}=E$. Then we have equivalence between :

1. $T_n(b) \to T(b) ~\forall b \in B$ and $\sup_{n}\|T_n\| < \infty$,
2. $T_n(x) \to T(x) ~\forall x \in E$.

I have been given the hint to use the Banach-Steinhaus theorem but I do not understand how. I know that 1 implies that $T$ is continuous but I do not know if this will help.

Thank you.