Understanding the proof if $f$ is continuous on $A \subset \mathbb{R}$ and $K \subseteq A$ is compact then $f(K)$ is compact

Question

I am trying to understand the proof of if $f$ is continuous on $A \subseteq \mathbb{R}$ and $K \subseteq A$ is compact $\implies$ $f(K)$ is compact. This is the proof:

Can someone please explain how $f(x) \in f(K)?$ It might be something simple (I'm guessing this has nothing to do with the continuity of $f$) but it is completely eluding me. Thanks!

Answer 1

The definition of $f(K)$ is the set of images of points in $K$ under $f$, i.e. $$f(K) := \{f(y) : y \in K\}.$$ Therefore, because $x \in K$, we have $f(x) \in f(K)$ by definition.