I am reading the proof of the theorem shown below (from Linear Functional Analysis by Rynne and Youngson). I can't figure out why the part I highlighted in red is true. I understand why $S$ is compact (it's the unit sphere), but I don't know why that implies that we can find a minimum amongst all $n$-tuples in $S$. Could someone explain that to me? Thanks!
Many have provided a good answer in the comments already, but I'll just spell out every detail here:
It is a standard fact (in topology) that the image of a compact set under a continuous map is compact, and in $\mathbb{R}^n$, compact is equivalent to closed and bounded. So, for the continuous map $f:S\to \mathbb{R}$, $f(S)$ is a closed and bounded set in $\mathbb{R}$. Since it is bounded, infimum (greatest lower bound) of $f(S)$ exists, which we'll call $m$. And $f(S)$ closed tells us that $m\in f(S)$, so there exists $\vec\mu$ such that $m=f(\vec\mu)$