Let $H$ be a Hilbert space and $p\in B(H)$ an idempotent on $H$, i.e. a continuous linear map satisfying $p^2 = p$. Is it true that $p(H)$ is a closed subspace of $H$?
Attempt:
Consider a sequence $p(h_n) \to h $. We show $h \in p(H)$. By continuity, $$p(h_n) = p^2(h_n) = p(p(h_n)) \to p(h)$$ but also $p(h_n) \to h$ so we obtain $h = p(h) \in p(H)$ and we are done.
Is this correct?
Yes, your proof is correct.
As an alternative approach, we could note that $p(H) = \ker(\operatorname{id}_H - p)$, and the kernel of any bounded linear map is closed.