I am working through a proof that every hilbert space has a orthogonal basis which lies dense in that Hilbert space. In the proof the following is done:
Let $v$ be a vector, and $E \subseteq I$ an arbitrary subset of the (possible uncountable) index set $I$ and consider $$ v_E = \sum_{i \in E} c_i(v) e_i $$ with $c_i(v) = \langle v, e_i \rangle$.
Then in the proof it is concluded that $\sum_{i \in E} |c_i(v)|^2 \le ||v||^2$, and so $$ \sum_{i \in I} |c_i(v)|^2 \le ||v||^2. (*) $$ And then it is said because of that just countable many $c_i(v)$ could be unequal to $0$ and that the series of the $|c_i(v)|^2$ converges.
I don't see why because of (*) the set $\{ c_i(v) | c_i(v) \ne 0 \}$ is countable?