Net convergence and norm-convergence in Hilbert spaces Let $\mathcal H$ be a Hilbert space which is not necessarily separable. Given a sequence of element $h_n$ indexed by $\mathbb N$, we say their sum converges in norm if the sequence $\{\sum_{n=1}^k h_n \}$ converges in norm. This is the usual notion of a convergent sum in a Hilbert space.
However, there is also the concept of convergence as a net, which applies also to uncountable sums. Given some index set $I$ and elements $h_i$, $i\in I$, we let $\mathcal F$ be the collection of all finite subsets of $I$, ordered by inclusion. This makes $\mathcal F$ into a directed. set. For each $F\in \mathcal F$, we define 
$$h_F = \sum\{ h_i : i\in F\}.$$
This is a finite sum, so it is an element of $\mathcal H$. The set of sums $h_F$ forms a net in $\mathcal H$. We say the sum over $I$ of the $h_i$ converges if the net converges. (See here for more information on nets.)
Given a set of elements of $\mathcal H$ indexed by $\mathbb N$, one naturally wonders about how net convergence and norm convergence relate. 
I would like to prove the following.

If $\{ h_n\}$ is a sequence in Hilbert space and $\sum\{ h_n: n\in
 \mathbb N\}$ converges to $h$ as a net, then $h_n$ converges to $h$ in
  norm. The converse is false.

The problem and definitions are taken from chapter 1, section 4 of Conway's A Course in Functional Analysis, second edition. 

If the sum converges to $h$ as a net, that means for every $\varepsilon > 0$, you have a finite set $F_\varepsilon \subset \mathbb{N}$ such that for every finite $F \supset F_\varepsilon$ you have
$$\left\lVert \sum_{n \in F} h_n - h\right\rVert < \varepsilon.$$
In particular, for all $N \geqslant \max F_\varepsilon$, you have
$$\left\lVert\sum_{n=1}^N h_n - h\right\rVert < \varepsilon,$$
that means $\sum h_n$ converges in norm.
Conversely, let $\mathcal{H} = \mathbb{R}$ and $h_n = \frac{(-1)^{n-1}}{n}$. It is well-known that the sequence of partial sums $s_k =\sum_{n=1}^k h_n$ converges in norm (to $\log 2$), but  for every finite set  $F_\varepsilon$, there exists a finite subset $F \supset F_\varepsilon$ with $$\sum_{n\in F} h_n > K$$ for every $K \in \mathbb{R}$.
