Clarification on analysis notes

Question

Someone would be so kind as to explain to me how the highlighted part came about? Why is the function Min used instead of Max?

Answer 1

Here's the trick: In the string of inequalities, there is the line $$ |u|^3+2|u|<|u|+2|u|. $$ In other words, $|u|^3<|u|$. This only happens when $0<|u|<1$. Therefore, one condition that will need to hold is $|u|<1$.

Since you also want $|f(x)-7|<3|u|<\epsilon$, you also need $|u|<\epsilon/3$.

Recapping, you know that you need both $|u|<1$ and $|u|<\epsilon/3$ to hold. So, you need $|u|$ to be smaller than the smallest of these two, which is the minimum of the two.