Uniform convergence preserves boundedness

Uniform convergence preserves boundedness: Let $E$ be a set and let $f_n:E\to\mathbb{R}$ be functions. If each $f_n$ is bounded on $E$ and $f_n\to f$ uniformly on $E$, then $f$ is bounded on $E$. Moreover, the family $\{f_n:n\in\mathbb{N}\}$ is uniformly bounded: there exists $M<\infty$ such that $|f_n(x)|\le M$ for all $n$ and all $x\in E$.

This is frequently used together with the supremum-norm characterization of uniform convergence.