Let $K \subset \mathbb R$ be compact, and let $(f_n)_{n=1}^{\infty}$ be a sequence of functions $f_n: K \to \mathbb R$. Suppose there exists a [continuous function](/page/Continuous%20Function) $g: K \to [0,\infty)$ such that
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\begin{align*}
|f_n(x)| \le g(x)
\end{align*}
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for every $n \in \mathbb N$ and every $x \in K$. Then $(f_n)_{n=1}^{\infty}$ is uniformly bounded on $K$; that is, there exists a constant $M \ge 0$ such that