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Suppose that you have an orthonormal basis $\{e_n\}$ in a Hilbert space such that $\sum \|e_n-x_n\| < 1$. Is this condition enough to prove that the closed span of $\{x_n\}$ is $H$?

My efforts to prove this have not led anywhere promising. I have tried showing that the only vector perpendicular to all of the $x_n$ would be $0$. Not sure which way I can proceed.

Does anyone have an idea how to approach this? Thank you.