You can't just drop the sum. You'd have to multiply by the cardinality of , which I don't think you want to do. It's no different, mathematically, from the sum
Your proof doesn't appear to use separability. I'm not sure off-hand how that can work, at the moment, but I have a feeling that's important. I do know that there's a theorem that says any orthornomal set in a separable Hilbert space is countable. Your set is not at the moment orthonormal, but it is pairwise orthogonal. You could simply run through your set and normalize each one to get an orthonormal set. What you don't know is if your set is a basis. If it were, you could write as a linear combination of elements of the set, and then I think your proof would be rather straight-forward. As it is, you'll have to do something else. Assuming you've orthonormalized your set, you do know the distance between any two elements in the set (think unit vectors).
So, while I don't know the proof solution, I can answer your question. Hope this helps.