Let $D$ be an operator defined by $D(c_n)_{n\in\mathbb{N}} = (a_n c_n)_{n\in\mathbb{N}}$. It's a well known fact that $D$ is well defined as an operator from $l^2(\mathbb{N})$ to $l^2(\mathbb{N})$ if $(a_n)_{n\in\mathbb{N}} \in l^\infty(\mathbb{N})$.

Does the converse hold, i.e. if $D\in L(l^2(\mathbb{N}))$ then $(a_n)_{n\in\mathbb{N}} \in l^\infty(\mathbb{N})$?

For me this seems like an easy to answer question, but I did not find any proofs or statements in the literature. I think I found a proof by myself but I want to be sure about that and I have the feeling that sequences like $(\log n)_{n\in\mathbb{N}}$ make problems.