Suppose we have an orthogonal family $\,\phi _{n}~~, n=1,2,...,$ in a Hilbert space, and $a_{n}$ is any sequence of real scalars such that $\,\sum_{i=1}^{\infty}\left| a_{i}\right|^{2}\langle\phi _{i},\phi _{i}\rangle \,$ converges.
How can we prove that $\,\sum_{i=1}^{\infty}a_{i}\phi _{i} \,$ is cauchy in H?
Here is my new attempt using the suggested method:
Let $\,g_{n} = \,\sum_{i=1}^{n}a_{i}\phi _{i} \,$, we must then prove that there exists $N$ such that $\,\lVert g_{n}-g_{m}\rVert<\epsilon\,$ for every $n,m>N$.
$$\,\lVert g_{n}-g_{m}\rVert^2=\langle g_{n}-g_{m},g_{n}-g_{m}\rangle$$ assuming $n<m$: $$\,\lVert g_{n}-g_{m}\rVert^2=\langle\, \sum_{i=n}^{m}a_{i}\phi _{i}\,,\sum_{j=n}^{m}a_{j}\phi _{j}\,\rangle=\sum_{i=n}^{m}a_{i}\langle \phi _{i},\sum_{j=n}^{m}a_{j}\phi _{j}\ \rangle $$
$$=\sum_{i=n}^{m}a_{i}\bigg[\sum_{j=n}^{m}a_{j}\langle \phi_{i} , \phi_{j} \rangle\bigg]=\sum_{i=n}^{m}\sum_{j=n}^{m}a_{i}a_{j}\langle \phi _{i},\phi _{j} \rangle$$
Using $\,\langle \phi _{i} ,\phi _{j}\rangle = 0 \,\,$when $i \neq j\,$, we end up with:$$\lVert g_{n}-g_{m}\rVert^2= \sum_{i=n}^{m}\left| a_{i} \right|^2\langle \phi_{i} , \phi_{i} \rangle$$
But I still can't extract the $N (\epsilon)$ when I impose:
$$\sum_{i=n}^{m}\left| a_{i} \right|^2\langle \phi_{i} , \phi_{i} \rangle < \epsilon$$
Do an explicit computation instead of using inequalities.
Use orthogonality to show that $\|g_n-g_m||^{2}=\sum\limits_{k=n+1}^m |a_k|^{2} \langle \phi_k, \phi_k \rangle$ for $n <m$. This sum tends to $0$ by hypothesis.