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.

• Could you make your question less verbose and a little more clear? I'm genuinely having a difficult time trying to really understand what your question is. – Daniel W. Farlow Feb 8 '15 at 5:03
• Edited. If this isn't clear enough, I can edit it again. Sorry about that, I would put an explicit example up but I'm having a tough time thinking one up. – user208786 Feb 8 '15 at 5:09
• I think that's ok, provided the failure implies that $Q$ is wrong but not just "not able to show that $Q$ is correct by induction". – velut luna Feb 8 '15 at 5:16