Let $X$ be a set, let $\mathbb{F}$ be either $\mathbb{R}$ or $\mathbb{C}$, and let $(f_k)_{k=1}^{\infty}$ be a sequence of functions $f_k:X\to\mathbb{F}$. If $(f_k)_{k=1}^{\infty}$ converges pointwise on $X$, then $(f_k)_{k=1}^{\infty}$ is pointwise bounded on $X$; that is, for every $x\in X$ there exists a constant $M_x<\infty$ such that

\begin{align*} |f_k(x)| \le M_x \end{align*}

for every $k\in\mathbb{N}$.