Proposition 5.2.8. Let $X$ be a compact uniform space, $Y$ be a uniform space, and $f \in C(X; Y)$, then $f \in UC(X; Y)$.
Proof. Let $V$ be an entourage of $Y$. For each $x \in X$, let $U_{x}$ be an entourage of $X$ such that $(f(y), f(z)) \in V$ for all $y, z \in (U_{x} \circ U_{x})(x)$. Since $X$ is compact, there exists $\seqf{x_j}\subset X$ such that $X = \bigcup_{j = 1}^{n}U_{x_j}(x_{j})$.
Let $U = \bigcap_{j = 1}^{n} U_{x_j}$, then for any $(x, y) \in U$, there exists $1 \le j \le n$ such that $x \in U_{x_j}(x_{j})$. In which case, $x, y \in (U_{x_j}\circ U_{x_j})(x_{j})$, so $(f(x), f(y)) \in V$.$\square$