Uniform convergence implies pointwise convergence

Corollary: Let $X$ be a set and let $(f_n)$ be functions $f_n:X\to Y$ into a metric space $(Y,d)$. If $f_n\to f$ uniformly on $X$, then $f_n\to f$ pointwise on $X$; i.e., for every $x\in X$, $f_n(x)\to f(x).$

Connection to parent theorem: Uniform convergence means $\sup_{x\in X} d(f_n(x),f(x))\to 0$. In particular, for each fixed $x$, $d(f_n(x),f(x))\le \sup_X d(f_n,f)\to 0$.