Proof by inductions often involve showing that Pn implies Pn+1. That is, that a statement's truth for n implies it's truth for n+1. That's what's being done her

Proof by inductions often involve showing that Pn implies Pn+1. That is, that a statement's truth for n implies it's truth for n+1. That's what's being done here, and it's a perfectly valid part of this type of proof.