Absolute Net Convergence Equivalent to Absolute Convergence

Theorem
Let $V$ be a Banach space.

Let $\sequence {v_n}_{n \mathop \in \N}$ be a sequence of elements in $V$.

Let $r \in \R_{\mathop \ge 0}$

Then the following two statements are equivalent:


 * $(1): \quad$ the generalized sum $\ds \sum \set {v_n: n \in \N}$ is absolutely net convergent to $r$
 * $(2): \quad$ the series $\ds \sum_{n \mathop = 1}^\infty v_n$ is absolutely convergent to $r$

Statement $(1)$ implies Statement $(2)$
Let the generalized sum $\ds \sum \set {v_n: n \in \N}$ be absolutely net convergent to $r$.

Statement $(2)$ implies Statement $(1)$
Let the series $\ds \sum_{n \mathop = 1}^\infty v_n$ be absolutely convergent to $r$.