Proving that if $\sum\|f_n-e_n\|^2< 1$, $\{f_n\}$ is a complete sequence
Solution 1:
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.