The constructive solution in case where $f_n$ are linearly independent.
Let $g_n$ be the orthonormal sequence obtained by Gram-Schmidt process from $f_n$. Construct inductively a sequence $\alpha_n$ in the following way:
- put $\alpha_0=1$
- given $\alpha_n$ for $n, put $0<\alpha_N\leq 2^{-N}$ other than $-\sum_{n.
Put $g=\sum_n \alpha_ng_n$, then $\langle g\vert f_N\rangle=\alpha_N\langle g_N\vert f_N\rangle+\sum_{n
In general, let $e_n$ be an orthonormal sequence obtained from $f_n$ by Gram-Schmidt process. Construct inductively sequence $\alpha_n$:
- $\alpha_0=1$
- Given $\alpha_n$ for $n, put $0<\alpha_N<2^{-N}$ such that for any $f_k$ in the linear span of $e_0,\ldots,e_N$, but not in the span of $e_0,\ldots,e_{N-1}$ we have $\alpha_N\neq-\sum_{n. These are only countably many points, so we can choose such an $\alpha_N$.
Similarly to the above, $g=\sum_n\alpha_ne_n$ will be okay (because the $e_n$ comes from Gram-Schmidt, any $f_n$ will be spanned by finitely many of them).