Can you apply the induction hypothesis to its outcome?
As stated, the reasoning does not make much sense to me.

You claim that, in your specific case, you have


$a &lt; b \wedge P(a) \Longrightarrow \exists c &lt; b$
$P(a) \wedge P(b) \Longrightarrow P(b)$


but these properties are trivial tautologies -- they tell nothing about your specific case.

In the first one, if $a &lt; b$, we already know $\exists c&lt;b$. We don't even need $P(a)$.

The second one is an instance of the tautology $p \land q \implies q$.

I'm not sure how these two properties help in the induction proof at all.