I'm claiming that this proof is using the fact that $u < m+1$, which isn't valid. Because for any finite set of positive integers $K$, $\sup K = \max K$. Which means that it would be impossible for the supremum of a set of positive integers to be less than another integer in the same set. That is, the number $m$ in the author's proof must be the next integer less than the supremum.
Proof
Firstly, for all $k \in K$, $k \leq \max K$. So $\max K$ is an upperbound for $K$.
Now, let $v \in \mathbb{R}$ be any number less than $\max K$. Define $\epsilon = \max K - v >0$. Choose $s_\epsilon = \frac{\max K+v}{2}$. Then
$$\max K - \epsilon = v < \frac{\max K+v}{2} = s_\epsilon $$
Proving that $\sup K = \max K$. That means the author's supremum $u$ in the proof is the maximum integer of the nonempty set $\mathbb{N}$. Subtracting $1$ from $u$ would provide the next lowest integer. There would indeed exist $m \in \mathbb{N}$ s.t. $u -1 < m$ but that $m$ would have to equal $\max{\mathbb{N}}$. So, in conclusion, $u \nless m +1$.
Please let me know what you think.
The proof in the book is a proof by contradiction. As such, it contains a sequence of false statements. False statements in the proof include:
The idea of a proof by contradiction is that by the time we get to the last false statement, we know it is false. As you correctly observed, it is impossible that $u < m+1,$ given the way $u$ and $m$ were constructed. But in the proof, all of the false statements (including the last one) were shown to be consequences of the first false statement. Now that we know the last statement is false, we know the first statement also is false. Since the first statement was the negation of the thing we wanted to prove, we know the thing we wanted to prove is true.