Let $\{e_n\}$ be a complete orthonormal sequence in an Hilbert space $H$ and let $\{f_n\}$ be an arbitrary sequence of elements in $H$ s.t $$\sum_{n=1}^\infty\|f_n-e_n\|^2<1$$Show that $\{f_n\}$is a complete sequence
First I thought simplifying the last inequality:$$1>\sum_{n=1}^\infty\|f_n-e_n\|^2=\sum_{n,k=1}^\infty|(e_n-f_n,e_k)|^2=\sum_{n,k=1}^\infty|\delta_{n,k}-(f_n,e_k)|^2$$but I was stuck there.
Incorrect Solution: Another option I thought about was using parsevall's generalized identity on $f_n$ means: $$0=(f_n,f)=\sum(f_n,e_n)\overline{(f,e_n)}$$ and since the inner product is non-negative, if $(f,e_n)=0$ then $f\equiv 0$ and we are done and otherwise $(f_n,e_n)=0$ which means that $\forall n\in\mathbb N,f_n=0$ and then by definition of inner product $\{0\}$ is indeed a complete sequence.
What's incorrect in the 'incorrect solution' and how can I use the fact about the inequality?
I take it by "complete" you mean that the $f_n$ have dense linear span or, equivalently, they have zero orthogonal complement. This we can see as follows. Suppose that $\langle g, f_n\rangle =0$ for all $n$. Then $$ |\langle g, e_n\rangle | =|\langle g, e_n-f_n\rangle | \le \|g\|\, \|e_n-f_n\| . $$ Take squares and sum over $n$. This yields $\|g\|^2 \le S \|g\|^2$ where $S<1$ is the sum from your assumption. Thus $g=0$, as desired.