Equicontinuity and dense sets lemma

Equicontinuity and dense sets lemma: Let $K$ be a compact metric space , and let $D\subseteq K$ be dense in $K$. Let $(f_n)$ be an equicontinuous sequence of functions $f_n:K\to\mathbb{R}$. If for every $x\in D$ the numerical sequence $(f_n(x))$ is Cauchy (equivalently, convergent), then $(f_n)$ is uniformly Cauchy on $K$.

Consequently, by the uniform Cauchy criterion , the sequence $(f_n)$ converges uniformly on $K$ (since $\mathbb{R}$ is complete), a key step in many proofs of Arzelà–Ascoli -type compactness results.