Let $X$ be a nonempty set. Let $(f_n)_{n\in\mathbb N}$ be a sequence of functions $f_n:X\to\mathbb R$, and let $f:X\to\mathbb R$ be a function such that $f_n(x)\to f(x)$ for every $x\in X$. Suppose that $(f_n)_{n\in\mathbb N}$ is uniformly bounded, in the sense that