I need to proof the following statements, having $ \quad X = \{x_n : \mathbb{N} \rightarrow \mathbb{C}\}$ and $p_j = max_{k \le j}|x(k)|$
1)$p_j$ is a countable family of seminorms which induces Hausdorff topology
2) $(X,\{p_j\})$ is a Frèchet space
For 1) I know that a countable family of seminorms induces Hausdorff topology if $\quad \forall x \in X-0 \quad \exists p \in \{p_j\} : p(x) > 0$. So of course any non-null sequence satisfies this condition.
But how should I prove 2)? First I should prove that it is a locally convex space, but I don't know where to start...