How can I prove that $f$ is continuous at $0$?

Question

Let $E$ and $F$ linear normed spaces, and consider a linear function $f:E\to F$ which satisfies for all $(x_n)\in E^{\mathbb{N}}$ such that $x_n\to 0$ then $f(x_n)$ is bounded in $F$. Then I have to prove that $f$ is continuous.

My approach: This is a second part of a problem. I have proved that: if $f$ satisfies $f(x+y)=f(x)+f(y)$ and is bounded in $B(0,1)$. Then $f$ is linear and continuous. This part is needed for the second? I'm trying to prove that is continuous at $0$ but without success.

Answer 1

I claim that $\exists M \geq 0$ such that $$ |f(x)| \leq M \quad\forall x\in E \text{ such that } \|x\| \leq 1 \qquad(\ast) $$ If not, then $\exists x_n \in E$ such that $\|x_n\| \leq 1$ and $$ |f(x_n)| > n^2 $$ Hence, take $$ y_n = x_n/n $$ Then, $y_n \to 0$ and $\{f(y_n)\}$ is unbounded.

Now can you prove that $f$ is continuous at $0$ from $(\ast)$?