$E$ is a $Banach$ space, $p$ is a seminorm on $E$ if $p$ satisfies $\sigma$-additivity, i.e. if $\sum_{n=1}^\infty x_n $ converges in $E$, then $$p(\sum_{n=1}^\infty x_n) \le \sum_{n=1}^\infty p(x_n) \le \infty$$ Show $p$ is continuous.
I know it's equivalent to that $\exists M \ge 0$, such that $p(x) \le M\|x\|.$
I made the following attempt:
$\forall x \in E\quad \forall (x_n), x_n \to x$. Let $y_1=x_1, y_n=x_n -x_{n-1},n \ge 2$.
Then $\sum_{n=1}^\infty y_n \to x,$
then $p(x)=p(\sum_{n=1}^\infty y_n) \le \sum_{n=1}^\infty p(y_n).$
But I couldn't find the relationship between $\sum_{n=1}^\infty p(y_n)$ and $\|x\|.$
