Extreme Value Theorem on a Compact Subset of a Metric Space

In the setting of The Real Numbers, with the Natural Numbers, Integers and Rationals Identified with Subsets of the Reals, and Completeness, let $(X,d)$ be a metric space, let $\mathcal{T}_{d}$ be the set of all $U\in\mathcal{P}(X)$ that are open in $(X,d)$, a topology on $X$ by The Open Subsets of a Metric Space Form a Topology (i), and let $K\subseteq X$ be nonempty and compact in $(X,\mathcal{T}_{d})$. Let $d_{\mathbb{R}}$ be the distance $d_{\mathbb{R}}$ on $\mathbb{R}$, a metric on $\mathbb{R}$ by The Absolute Value Metric on the Real Numbers (i), and let $f:K\to\mathbb{R}$ be continuous on $K$ as a map from $(X,d)$ to $(\mathbb{R},d_{\mathbb{R}})$.

(i) There exist $x_{\min},x_{\max}\in K$ such that

$$f(x_{\min})\le f(x)\le f(x_{\max})\qquad\text{for every }x\in K .$$