Proposition 8.1.3.label Let $\seq{(X_n, d_n)}$ be metrisable spaces, then $\prod_{n \in \natp}X_{n}$ is also metrisable.
Proof. For each $n \in \natp$, let
\[d_{n}': \braks{\prod_{n \in \natp}X_n}^{2} \to [0, \infty] \quad (x, y) \mapsto d_{n}(\pi_{n}(x), \pi_{n}(y))\]
then $d_{n}'$ is a pseudometric on $X$, and $\bracsn{d_n'}_{1}^{\infty}$ induces the product uniformity on $\prod_{n \in \natp}X_{n}$. By Theorem 6.3.10, $\prod_{n \in \natp}X_{n}$ is also metrisable.$\square$
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