Uniform convergence implies pointwise convergence

Uniform convergence implies pointwise convergence: Let $E$ be a set and let $f_n:E\to\mathbb{R}$ (or $\mathbb{C}$) be functions. If $f_n\to f$ uniformly on $E$, then $f_n\to f$ pointwise on $E$; that is, for every $x\in E$ we have $f_n(x)\to f(x)$.

The converse generally fails: pointwise convergence alone is typically too weak to preserve analytic properties such as continuity.