Let $(K,d)$ be a compact [metric space](/page/Metric%20Space), let $\mathbb{F}\in\{\mathbb{R},\mathbb{C}\}$, and let $(f_k)_{k=1}^{\infty}$ be an equicontinuous sequence of functions $f_k:K\to\mathbb{F}$. Suppose that $(f_k)_{k=1}^{\infty}$ is pointwise bounded on $K$, meaning that for every $x\in K$ there exists $B_x<\infty$ such that $|f_k(x)|\le B_x$ for every $k\in\mathbb{N}$. Then $(f_k)_{k=1}^{\infty}$ is uniformly bounded on $K$: there exists $M<\infty$ such that $|f_k(x)|\le M$ for every $x\in K$ and every $k\in\mathbb{N}$.