I have a question about general proof writing. Unfortunately I don't have an explicit example to illustrate my question, but here it is:
Let's say you wanted to prove something by contradiction. Begin the proof as you normally would, if the statement to be proven is $P$, assume ~$P$. Now, instead of showing that this leads to a contradiction directly, choose some mathematical statement $Q$ that is known to be true and can be proven using induction. Now begin to prove $Q$ using induction, but in the inductive step, show that ~$P$ leads to failure of the inductive step. Would this be an acceptable way to prove something?
Thanks for any input.