In which conditions the limit of an infinite sum is equal to the infinite sum of the limits?

Question

Let $X \neq \varnothing$ be a set, $\{x_k\}_{k\in \Bbb{N}} \subset X$ and $\{f_n : X \hookrightarrow[0,\infty]\}_{n\in \Bbb{N}}$ such that $\forall n \in \Bbb{N} \, \forall x \in X : f_n(x) \leq f_{n+1}(x)$.

I want to know in which conditions $$\sum_{k\in \Bbb{N}} \lim_{m \rightarrow \infty} f_m(x_k) = \lim_{m \rightarrow \infty} \sum_{k\in \Bbb{N}} f_m(x_k)$$

My attempt guides me to the following by knowing that the limit of a increasing sequence is greater than all the elements of the sequence: $$\forall n \in \Bbb{N} \, \forall x \in X : f_n(x) \leq f_{n+1}(x) \Rightarrow \forall n,k \in \Bbb{N} : f_n(x_k) \leq f_{n+1}(x_k) \Rightarrow \forall n,k \in \Bbb{N}:f_n(x_k) \leq \lim_{m \rightarrow \infty} f_m(x_k) \Rightarrow \forall n \in \Bbb{N}:\sum_{k\in \Bbb{N}} f_n(x_k) \leq \sum_{k\in \Bbb{N}} \lim_{m \rightarrow \infty} f_m(x_k) \Rightarrow \lim_{m \rightarrow \infty} \sum_{k\in \Bbb{N}} f_m(x_k) \leq \sum_{k\in \Bbb{N}} \lim_{m \rightarrow \infty} f_m(x_k) $$

I don't know if the other inequality is true, neither which conditions should make it true. Any possible answers would be appreciated.

Answer 1

I apologize for my previous mistakes. I'm also currently learning real analysis and my knowledge is quite fragmented.

For simplicity I will write $f_m(k)$ instead of $f_m(x_k)$.Tannery's theorem in my previous response does not make full use of the conditions of monotonicity and non-negativeness of $f_m(x)$. Instead, this is exactly the statement of Monotone convergence theorem:

If for all $m,k\in\mathbb N, f_m(k)\in[0,+\infty]$ and $f_m(k)\le f_{m+1}(k)$, then $$\sum_{k\in \mathbb N}\lim_{m\to\infty}f_m(k)=\lim_{m\to\infty}\sum_{k\in \mathbb N}f_m(k)$$