A Closed Interval is Sequentially Compact in the Real Line

Let $\mathbb{R}$ denote the real numbers, with the order $\le$ of its ordered field structure, and let $(\mathbb{R},d_{\mathbb{R}})$ be the real line, that is, $\mathbb{R}$ equipped with the absolute value metric. Let $a,b\in\mathbb{R}$ satisfy $a\le b$, and let $[a,b]$ be the closed interval determined by $a$ and $b$.

Then $[a,b]$ is sequentially compact in $(\mathbb{R},d_{\mathbb{R}})$.