Local Uniform Convergence Implies Compact Convergence

Theorem
Let $X$ be a topological space.

Let $M$ be a metric space.

Let $(f_n)$ be a sequence of mappings $f_n : X\to M$.

Let $f_n$ converge locally uniformly to $f:X\to M$.

Then $f_n$ converges compactly to $f$.

Also see

 * Compact Convergence Implies Local Uniform Convergence if Weakly Locally Compact
 * Equivalence of Local Uniform Convergence and Compact Convergence