Local Uniform Convergence Implies Compact Convergence

Theorem

Let $X$ be a topological space.

Let $M$ be a metric space.

Let $\sequence {f_n}$ 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 $f$.

Proof

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Also see

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