Let $E \subset \mathbb{R}$ be nonempty, and let $(f_n)_{n \in \mathbb{N}}$ be a sequence of bounded functions $f_n: E \to \mathbb{R}$. For each $n \in \mathbb{N}$, define the ordinary supremum norm on $E$ by
Then $(f_n)_{n \in \mathbb{N}}$ is uniformly bounded on $E$ if and only if the numerical sequence $(\|f_n\|_{\infty,E})_{n \in \mathbb{N}}$ is bounded above.