The problem here is that if you tried to prove, then by adding to both sides (which becomes necessary because of induction) we get, , but we cannot continue any longer because and induction breaks down here.

That was already addressed on top. It is necessary because of induction.2. Why should we add to both sides ?

Over here we might use instead of because we want to show this holds for . In order for this statement to make any sense we need to realize that is a problem for then we have a zero denominator. There may be other reasons but this is the first one that I thought of.3. In a similar problem, having to prove

we shall prove

Why choose and then add to both sides ?