$f:X\rightarrow Y$ continuous $X$ compact uniform space, then uniformly continuous? If $(X,\mathfrak D)$ and $(Y,\mathfrak C)$ are uniform spaces, $X$ compact and $f:X\rightarrow Y$ continuous, why $f$ is uniformly continuous? 
 A: Yes. Every continuous function on a compact uniform space is uniformly continuous. [Isbell J.R. Uniform spaces (MSM012, AMS, 1964)] 
A: I suppose $\mathfrak D$ and $\mathfrak C$ are the diagonal uniformities of $X$ and $Y$ respectively.
Let $E\in \mathfrak C$, let us find $D\in \mathfrak D$ such that $(x,y)\in D \Rightarrow (f(x),f(y))\in E$.
Choose $F\in \mathfrak C$ with $F\circ F^{-1}\subseteq E$ $(1)$.
Since $f$ is continuous, choose for each $x\in X$, $D_x\in \mathfrak D$ such that $f(D_x[x])\subseteq F[x]$ $(2)$. For each $x\in X$ pick $D_x'\in \mathfrak D$ with $D_x'\circ D'^{-1}_x\subseteq D_x$ $(3)$. Pick $x_0,\ldots, x_n\in X$ with $X=\bigcup_{i=0}^n D'_{x_i}[x_i]$. Put $D=\bigcap_{i=0}^n D'_{x}$,then $D\in\mathfrak D$. 
We claim that $(x,y)\in D \Rightarrow (f(x),f(y))\in E$. 
Let $(x,y)\in D$, since $X=\bigcup_{i=0}^n D'_{x_i}[x_i]$ there is some $i\leq m$ with $x\in D_{x_i}[x_i]$, that is, $(x_i,x)\in D'_{x_i}$, but in particular $(x,y)\in D'_{x_i}$ , thus by $(3)$ $(x_i,y)\in D_{x_i}$, but clearly $D_{x_i}'\subseteq D_{x_i}$ so that $(x_i,y),(x_i,x)\in D_{x_i}$, i.e., $x,y\in D_{x_i}[x_i]$, then $f(x),f(y)\in F[f(x_i)]$; by $(2)$, that is, $(f(x_i),f(x)),(f(x_i),f(y))\in F$, thus by $(1)$ we obtain $(f(x),f(y))\in E$ as desired.
