I have a Hilbert space $H$ and a base $(e_n)_{n=1}^\infty$ and a ON-sequence $(f_n)_{n=1}^\infty$.
Given $$ \sum_{n=1}^\infty ||e_n - f_n||^2 < 1 $$ show that $(f_n)_{n=1}^\infty$ is a base.
My work: It is straight forward to rewrite the sum $$ \sum_{n=1}^\infty (1 - \langle e_n, f_n\rangle) = \sum_{n=1}^\infty \langle e_n, e_n - f_n\rangle) < \frac12 $$ and it also holds for each $n$ $$ \langle e_n, f_n\rangle > \frac12 $$
I then I have tried to show by contradiction by assuming there is a non-zero vector $a$ that is orthogonal to $f_n$: Assume there is a vector $a$ s.t $||a|| = 1$ and $\langle a, f_n\rangle = 0$ for all $n$. $$ \langle a, f_n \rangle = \langle \sum_{m=1}^\infty\langle a, e_m\rangle e_m, \sum_{k=1}^\infty\langle f_n, e_k \rangle e_k \rangle = \sum_{m=1}^\infty \langle a, e_m\rangle \langle f_n, e_m \rangle = ... $$ and I want to somehow show that this is $> 0$ for a contradiction, based on the given inequality.
Am I no the right track? I'm just treading water right now and end up rewriting all the expressions in different ways without progress.
Hints are appreciated (note the homework tag)
Hint: $$\left|\langle a,e_n \rangle\right| = \left|\langle a, e_n - f_n \rangle \right| \le \|a\| \|e_n - f_n\|$$ Now use the given inequality...