locally convex space generated by a countable family of seminorms is metrizable

Question

Let $X$ be a locally convex Hausdorff topological space, whose topology if generated by the countable family of seminorms $\{p_i:\space i\in\mathbb{N}\}$. I'd like to prove that $X$ is metrizable.
So, let $$d(x,y)=\sum_{k=1}^{\infty}\frac{1}{2^k}\frac{p_k(x-y)}{1+p_k(x-y)}.$$ It is easy to see that $d$ is a metric indeed. I'd like to prove that this is a one we need, that generates the original convergence. Is that true by the way?

Let $x_n\to x$ in the introduced metric. Then for all $k$ we have $\frac{1}{2^k}\frac{p_k(x_n-x)}{1+p_k(x_n-x)}\le d(x_n,x)$ and thus $p_k(x_n-x)\to 0$ for each $k$. The last means the convergence in the original locally convex topology.

How to do the converse? So, suppose $p_k(x_n-x)\to 0$ ($n\to \infty$) for each $k$. How can I prove that $d(x_n,x)\to 0$. Althouhg it seems to be an elementary mathematical analysis, could you tell me wheather the metric I inroduce suits us and how to prove that if yes?