Net Convergence Equivalent to Absolute Convergence

Theorem
Let $V$ be a Banach space.

Let $\left({v_n}\right)_{n \in \N}$ be a sequence of elements in $V$.

Then the following two statements are equivalent:


 * $(1): \qquad \displaystyle \sum_{n=1}^\infty \left\Vert{v_n}\right\Vert$ converges (absolute convergence)
 * $(2): \qquad \displaystyle \sum \left\{{v_n: n \in \N}\right\}$ converges (generalised or net convergence)