The countable product of Fréchet spaces is a Fréchet space

Question

Let $\{E_n \; ; \; n \in \mathbb{N}\}$ be a family of Fréchet spaces. I want to prove that the product
$$E:= \prod_{n=1}^{\infty} E_n$$
is a Fréchet space, that is, $E$ is metrizable (Hausdorff space and admits a countable basis of neighborhoods of $0\in E$), complete and locally convex (admits a basis of neighborhoods of $0\in E$ consisting of convex sets).

I already know that $ E $ is Hausdorff, locally convex and complete space. I don't know how to prove that $ E $ is metrizable. How to proceed?

Answer 1

Let $d_n$ be one metric in each $E_n$. Then $$d(\tilde{x}, \tilde{y}):= \sum_{n=1}^{+\infty} \frac{d_n(x_n,y_n)}{1+d_n(x_n,y_n)}\frac{1}{2^n} $$ where $\tilde{x}, \tilde{y}\in E$, is a metric in $E$.