The following is from chapter 1, section 4 of Conway's A Course in Functional Analysis, second edition: I tried a lot but have no idea how to solve it :
If ${\{h_n}\}$ is a sequence in a Hilbert space and $ \sum_{n =1}^{\infty} \left\lVert h_n \right\rVert < \infty$ show that $ \sum {\{h_n : n \in F}\} $ converges as a net.
Please see details of definition and similar problem from the book in here.
A hint also would be much appreciated.
Here's an idea:
note that from the finiteness of the sum $\sum_{n\in\mathbb{N}}\|h_n\|$ it is true that for every $\varepsilon>0$ there exists a natural number $N_{\varepsilon}$ such that \begin{equation*} \sum_{n\geq N_\varepsilon}\|h_n\|< \varepsilon. \end{equation*}
Now, to prove the claim it suffices to show that the sequence is Cauchy. For every $\varepsilon>0$ there is a finite set $S:=\{1,\dots,N_\varepsilon\}$ such that for every couple of finite sets $F,F'\supseteq S$ we have \begin{equation*} \left\| \sum_{n\in F}h_n - \sum_{m\in F'}h_m \right\| = \left\| \sum_{n\in F \setminus (F \cap F')}h_n - \sum_{n \in F' \setminus (F \cap F')} h_n\right\| \leq \sum_{n\geq N_\varepsilon} \|h_n\| < \varepsilon, \end{equation*}