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I have trouble with a little part of a proof in my textbook.

Consider a holomorphic function $f$ on a open set $G$. Let $\partial K(a,r)$ be the boundary of a disc contained in $G$ with center a and radius r.

My question is:

How can I deduce that $\sup \lbrace \vert f(w) \vert \vert w\in\partial K(a,r) \rbrace <\infty$. My textbook uses the argument that the statement is true, because $\partial K(a,r)$ is bounded and closed. But where does this come from? A theorem?

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Holomorphic functions are in particular continuous, and closed bounded subsets of $\mathbb{C}$ are compact. A continuous function on a compact domain is bounded.

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