Uniform limit of continuous functions is continuous

Corollary: Let $(X,d)$ be a metric space and let $f_n:X\to\mathbb{R}$ be continuous for all $n$. If $f_n\to f$ uniformly on $X$, then $f$ is continuous on $X$.

Connection to parent theorem: This is exactly the uniform limit theorem for continuity, often restated as a corollary once the theorem has been proved.