Application of the reverse Fatou's lemma for limits in the continuum

Question

Let $(f_{\delta})_{\delta \in [0,1)}$ be functions with $f_\delta : \mathbb{N} \mapsto [0, \infty)$, such that there exists $g$ satisfying $f_\delta \leq g$ for any $\delta \in [0,1)$ and $\sum_{n \in \mathbb{N}} g(n) < \infty$. Does the following inequality hold? $$ \limsup_{ \delta \rightarrow 0} \sum\limits_{n \in \mathbb{N}} f_{\delta}(n) \leq \sum\limits_{n \in \mathbb{N}} \limsup_{ \delta \rightarrow 0} f_{ \delta}(n). $$ I would like to use the reverse Fatou's lemma to justify it, my problem is that the limit $\delta \rightarrow 0$ is not a sequence of integer values. Is in this case the inequality still justified?