Let $\{e_n\}$ and $\{f_n\}$ be orthonormal sequences in a Hilbert space $H$ with $\|e_n\|=\|f_n\|=1$ and $$\sum_{n=1}^{\infty}\|e_n-f_n\|^2<\infty.$$ Prove or disprove: If $\{e_n\}$ is complete, then $\{f_n\}$ is also complete.
Actually if the condition is changed into $\sum_{n=1}^{\infty}\|e_n-f_n\|^2<1$, then the conclusion holds:
If $\{f_n\}$ is not complete, then there exists a nonzero $x \in \{f_n\}^{\perp}$. Since $\{e_n\}$ is complete, then $$\|x\|^2=\sum_{n=1}^{\infty}|(x,e_n)|^2=\sum_{n=1}^{\infty}|(x,e_n-f_n)|^2 \leq \|x\|^2\sum_{n=1}^{\infty}\|e_n-f_n\|^2,$$ which leads to a contradiction.
But now the method doesn't work so I am stuck. Thanks for any help!