Suppose $f:S\to T$ which is continuous and consider $A\subset S$ which is compact, is $f$ is uniformly continuous on A?
I have tried to prove it by considering a finite subcover of $A$ say $\bigcup B(x_i,r_i)$ and I want to show that for any $\epsilon$-ball of any element of $f(A)$ there existd a uniform $\delta$, such that any $\delta$-ball of any point in $A$ would be mapped into the $\epsilon$-ball. So take a small enough $\epsilon$. By continuity, there is a uniform $\delta$ such that $f(B_{\delta})\subset B_{\epsilon}$. But here comes the problem: I am not sure how to get a uniform $\delta$ such that $f(B_{\delta})\subset B_{\epsilon}$ for any point in $A$. Any idea to get this uniform $\delta$?