Misunderstanding of the word "span"?

Question

Let $\mathscr{H}$ be a separable Hilbert space with orthonormal basis $\{\psi_{n}\}_{n\in \mathbb{N}}$. I once read in a paper that $\{\psi_{0}\}^{\perp} = \text{span}\{\psi_{1},\psi_{2},...\}$, but I do not see how this is true. Because $\text{span}\{\varphi_{0}\}$ is closed, then $\mathscr{H} = (\text{span}\{\varphi_{0}\})\oplus \{\varphi_{0}\}^{\perp}$ holds. Moreover, $\{\psi_{1},\psi_{2},...\}$ is an orthonormal basis for $\{\varphi_{0}\}^{\perp}$, so that $\text{span}\{\psi_{1},\psi_{2},...\}$ is dense in $\{\varphi_{0}\}^{\perp}$. Hence, every $\psi \in \{\varphi_{0}\}^{\perp}$ can be written as a sequence of elements in $\text{span}\{\psi_{1},\psi_{2},...\}$ but it does not imply that $\psi$ itself is an element of the latter. Am I missing something? Maybe the paper was using the word span to mean: $$\psi = \sum_{n=1}^{\infty}\langle \psi_{n},\psi\rangle \psi_{n}?$$ This is not why the word span mean typically, tho.