Let $a,b\in\mathbb{R}$ satisfy $a<b$. For each $n\in\mathbb{N}$, let $f_n:[a,b]\to\mathbb{R}$ be continuous. Suppose there exists a function $f:[a,b]\to\mathbb{R}$ such that $f_n\to f$ uniformly on $[a,b]$. Then $f:[a,b]\to\mathbb{R}$ is continuous, and