Proof of Closed Interval $[a,b]$ is Compact in $\mathbb{R}$

By A Closed Interval is Sequentially Compact in the Real Line the interval $[a,b]$ is sequentially compact in the metric space $(\mathbb{R},d_{\mathbb{R}})$.

Applying Compactness and Sequential Compactness Agree for Subsets of a Metric Space to the metric space $(\mathbb{R},d_{\mathbb{R}})$ and the subset $[a,b]$, we conclude that $[a,b]$ is compact in $(\mathbb{R},\mathcal{T}_{d_{\mathbb{R}}})$.

By The Euclidean Distance on the Real Line is the Absolute Value Metric the metric $d_{\mathbb{R}}$ and the Euclidean distance on $\mathbb{R}^{1}$, identified with $\mathbb{R}$, are the same function, and a subset of $\mathbb{R}$ is compact with respect to the topology of one exactly when it is compact with respect to the topology of the other. Hence the conclusion holds equally for the topology determined by the Euclidean distance.