Let $\mathbb{K}$ denote either $\mathbb{R}$ or $\mathbb{C}$. Let $X$ be a [countable set](/page/Countable%20Set), and let $(f_k)_{k=1}^{\infty}$ be a sequence of functions
paragraph
admin
\begin{align*}
f_k:X \to \mathbb{K}.
\end{align*}
latex_env
admin
Assume that $(f_k)_{k=1}^{\infty}$ is pointwise bounded on $X$, meaning that for every $x \in X$ there exists a constant $M_x < \infty$ such that
paragraph
admin
\begin{align*}
|f_k(x)| \leq M_x
\end{align*}
latex_env
admin
for every $k \in \mathbb{N}$. Then there exists a strictly increasing sequence of indices $(k_j)_{j=1}^{\infty}$ such that the subsequence $(f_{k_j})_{j=1}^{\infty}$ converges pointwise on $X$.