[Math] reference for “X compact <=> C_b(X) separable” (X metric space)

Question

I know (and am able to prove via Stone-Čech compactification) that the following is correct:

Theorem: A metric space is compact if and only if its space of bounded, continuous, real-valued functions is separable in the uniform topology.

I use it in a paper for readers who are presumably not familiar with this kind of topology, so I cannot call it "obvious" or "well-known".
I would be thankful for a name and/or good reference to cite this theorem!

Answer 1

The result does appear in Dunford/Schwartz, Linear Operators Part I (page 437), but is only stated as an exercise.

Edit after @JosephVanName' comment: Conway's Functional Analysis has the result for completely regular spaces as Theorem 6.6 (page 140).