I realized I was wrong about my memory of a counterexample, and I think I have a solution, but it would be nice if someone could confirm that my reasoning makes sense.
1. Since L2 is complete, it suffices to show the sequence is Cauchy and that the sequence converges to zero in norm.
2. By orthonormality of the f's, the sequence is Cauchy (see below for boring calculations).
3. By similar calculations to those used in step 2, the sequence converges to 0 in norm.
Calculations for 2: