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Let $H$ be a Hilbert Space and $\{e_n\}_{n\in\mathbb{N}}$ an orthonormal basis. Now let $(x_n)$ be a sequence in $H$ satisfying

$$\sum_{n=1}^{\infty}||x_n-e_n||^2<1.$$

Prove that $\operatorname{lin}\{x_n:n\in\mathbb{N}\}$ is dense in $H$.

I don't know even how to start this problem...

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We have to show that if $y$ is such that $\langle y,x_n\rangle=0$ for each $n$, then $y=0$. For such a $y$, $$\langle y,e_n\rangle^2=\langle y,e_n-x_n\rangle^2\leqslant \lVert y\rVert^2\lVert x_n-e_n\rVert^2.$$ Now sum over $n$ and use Bessel's equality.

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