The proof is by induction on $n$.
By decreasing induction on $p$, ......
We show that ...... by reverse induction on $i$, starting at $i=n$ and working down to $i=0$.
We proceed by induction.
If we apply induction to (3) we see that ......
For the base step of the induction, consider a vertex $t$ in $A$.
We may require that the point $P$ lie in one of the trees constructed before or during the $i$th stage of the induction. [Note the subjunctive lie.]
Induction shows that if ......
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