Convergence of a series $\sum\limits_{\alpha \in \Lambda} x_{\alpha}$ of positive real numbers indexed by an uncountable set $\Lambda.$

Question

Let $\Lambda$ be a indexing set (not necessarily countable) and let $\{x_{\alpha}\}_{\alpha \in \Lambda}$ be a collection of non-negative real numbers. Then our instructor told that the convergence of the series $\sum\limits_{\alpha \in \Lambda} x_{\alpha}$ is equivalent to the convergence $$\lim\limits_{L\ \uparrow} \sum\limits_{\alpha \in L \subseteq \Lambda} x_{\alpha},$$ where $L$ is a net of finite subsets of $\Lambda.$

I don't understand whether it is a definition or a consequence. Every small help would be greatly appreciated.

Thanks for your time.