I am reading the book Introduction to Set Theory by Thomas & Karel, 3rd edition.
And I am stuck on the proof of Theorem 2.2 in chapter 3 about natural numbers. Specifically, part iii of the proof. It says that: $\lt$ is a linear order of $\Bbb N$ (where $\lt$ is the strict ordering on $\Bbb N$).
The proof proceeds by induction on $n$ for the statement
$\forall m,n \in \Bbb N$ either $m \lt n$ or $m=n$ or $n \lt m$.
The case $0$ is simple.
Then the induction step supposes that:
$\forall m,n \in \Bbb N$ either $m \lt n$ or $m=n$ or $n \lt m$
Here the authors go step by step for each part, namely they prove that $m \lt n+1$ then $m=n+1$ then the third part with which I have a problem of understanding. Namely they say, and I quote
Finally, if $n \lt m$, we would like to conclude that $n+1 \leq m$. This would show that for all $m,n \in \Bbb N$, either $m \lt n+1$ or $m= n+1$ or $n+1 \lt m$
And then they go by induction again on $m$ to prove that
if $n \lt m$, we would like to conclude that $n+1 \leq m$.
The thing I am not getting is why "we would like to conclude that $n+1 \leq m$." I would expect to aim for "we would like to conclude that $n+1 \lt m$" which concludes the main induction step. I.e I don't see why he used $\leq$ instead of $\lt$, or why does this work for proving the main induction.
Thanks in advance!