Compact Convergence Implies Local Uniform Convergence if Weakly Locally Compact

Theorem
Let $X$ be a locally compact 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 compactly to $f:X\to M$.

Then $f_n$ converges locally uniformly to $f$.

Proof
Because $X$ is locally compact, each point of $X$ has a compact neighborhood.

Because $f_n\to f$ compactly, each point of $X$ has a neighborhood on which $f_n\to f$ uniformly.

Thus $f_n$ converges locally uniformly to $f$.

Also see

 * Local Uniform Convergence Implies Compact Convergence