I am studying real analysis by my own and stumble into the following theorem, that Bartle's proves in its appendix, however, I do not understand the case 2. Specifically I do not understand why does he define $h_1$, and the induction hypothesis is not supposed to be that there is not induction from $ N_m \rightarrow N_k$, so how does it implies that $h_1$ is not an injection. Can somebody please clarify this part, or explain it thoroughly, for me it is not easily seen as the book states. Thanks
Here is a few ideas.
$h_1$ is well defined because $h$ is well defined and no element in $\mathbb N_{m-1}$ is being sent to $k+1$.
$h_1$ is not an injection because $m > k + 1\implies \color{red} {m -1 > k}$ and by the induction step $h_1 : \mathbb N _{m-1} \to \mathbb N_k$ can't be an injection, notice in this case we are assuming $h(p) = k+1 $.
Finally, $h$ is not an injection because we have seen that $h_1$ isn't, and we know that $h(p) = k+1$. Notice that $\mathbb N_{m-1} \cup \{p\} = \mathbb N_m$ and $\mathbb N_k \cup \{k+1\} = \mathbb N_{k+1}$.
Well, the induction hypothesis is actually that there's no injection from $\mathbb{N}_m$ into $\mathbb{N}_k$.
