The Continuous Image of a Compact Subset of a Metric Space is Compact

In the setting of Metric Spaces: Balls, the Metric Topology, and the Real and Complex Numbers as Metric Spaces and Routine Facts, Continued: Typed Arithmetic, Maps and Predicativity, let $(X,d)$ and $(Y,\rho)$ be metric spaces.

(i) Continuous images. Let $K\subseteq X$ be compact in $(X,\mathcal{T}_{d})$, and let $f:K\to Y$ be a function that is continuous on $K$. Then the image

$$f(K)=\{f(x):x\in K\}$$

is compact in $(Y,\mathcal{T}_{\rho})$.