Exercise 1, § 2

Let $X$ be a Hausdorff uniform space and let $(f_i)$ be a family of pseudometrics on $X$ which define the uniformity of $X$. Let $X_i$ be the metric space associated with the uniform space obtained by endowing $X$ with the structure defined by the single pseudometric $f_i$ (no. 1). Show that $X$ is isomorphic to a subspace of the product uniform space $\prod_i X_i$.