Compactness argument in SVD existence proof

We are interested in the compactness of the subset $S = \{v\mid ||v|| = 1\}\subseteq \Bbb C^n$. Compactness is relevant because among those vectors, $||Av||_2 \leq \sigma_1$, and compactness is used to guarantee the existence of a vector $v_1$ such that there is equality. In other words, the function $f:S \to \Bbb R$ given by $f(v) = ||Av||_2$ has sup $\sigma_1$ by definition of operator norm, and compactness guarantees that it is actually a max.

It depends on how the induced matrix norm was defined. I don't have the book handy, but I expect some pages earlier the authors to have put $$\|A\|_2=\sup_{\|v\|_2=1}\|Av\|_2.$$ Note that a function's sup is not always achieved (think of $1-\exp x$ ($x$ real) and $1$). Compactness (for your purposes here) is a quick way of saying that the sup is achieved by a vector $v$, i.e., there is a specific vector $v_1$ which satisfies $$\|v_1\|_2=1\text{ and }\|Av_1\|=\|A\|_2.$$