Prove that $f(A)$ is a separable subset

Question

Let $f: (M,d) \rightarrow (N,\rho)$ be continuous and let $A$ be a separable subset of M. Prove that $f(A)$ is separable.

My idea is to take countable dense subset of A and then construct an open ball for its points and use the definition of continuity in a metric space to conclude that $f(A)$ is separable.

Kindly help me to provide a rigorous proof.

Thanks!