Metrizable implies compactly generated

This article gives the statement and possibly, proof, of an implication relation between two topological space properties. That is, it states that every topological space satisfying the first topological space property (i.e., metrizable space) must also satisfy the second topological space property (i.e., compactly generated space)
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Statement

Any metrizable space is a compactly generated space.

Facts used

1. Metrizable implies first-countable
2. First-countable implies compactly generated

Proof

The proof follows from fact (1) and fact (2).