Is every compact metric space hereditarily separable?

Question

Let $X$ be a compact metric space. I see why all open and closed subsets of $X$ are separable. But is every subset of $X$ necessarily separable?

EDIT: Since $X$ is separable metric, it embeds into the Hilbert cube $[0,1]^\omega$, which is hereditarily separable, right? And so $X$ is also hereditarily separable.

Answer 1

Yes: a compact metric space is second countable, and second countability is hereditary and implies separability.