Why is the image of a compact operator separable? Let $A$ and $B$ be normed vector spaces and let $S\in \mathscr{K}(A,B)$ be a compact operator.

Question: How does it follow that the image of $S$ is separable?

Thanks for the help.

The space is a countable union of balls centered in zero:
$$A = \bigcup_{n\in N}B(0,n).$$
The image of $B(0,n)$ is precompact, therefore, separable. Countable union of separable sets is separable.
