On pg 45 of Baby Rudin we have:
22. A metric space is called separable if it contains a countable dense subset.
24. Let $X$ be a metric space in which every infinite subset has a limit point. Prove that $X$ is separable.
25. Prove that every compact metric space $K$ ... is separable.
Since "countable" is defined in Rudin to exclusively mean "infinite and countable", then clearly the results don't hold for finite metric spaces, since that would imply an infinite set is a subset of a finite one.
What am I misunderstanding?