When I'm proving some problems, am I allowed to use two different proof technique to prove a problem?
For instance, let's say I decided to do proof by contradiction on certain algorithm. Since algorithm has a pattern, I am thinking of using induction as well within contradiction proof.
So it would look like following:
Proof by contradiction
Prove base case of induction is false
Prove inductive step of induction is false.
Of course you can combine several different proof techniques within a larger proof.
But the way you do combine a prof by contradiction with a proof by induction makes little sense.
Say that you want to prove that $P(n)$ for all $n$. Induction would seem to be a good option, so you could try and prove the base and step, possibly using a proof by contradiction for each or both. So sure, using a proof by contradiction within the parts of a proof by induction could certainly make sense.
But you are doing something quite different. You are doing a proof by contradiction on the whole thing. But that means you have to show that if we do not have $P(n)$ for all $n$, i.e. that if there is some $n$ for which we do not have $P(n)$, then we get a contradiction.
But you certainly do not show that by showing that the base case for induction does not hold. So what if it does not hold? How is that a contradiction? Indeed, if it does not hold for the base case, then you certainly will not be able to prove that $P(n)$ for all $n$, since that is apparently false.
Similarly, if the step not hold, you have not established any kind of contradiction. All you have shown is that $P(n+1)$ does not follow from $P(n)$. So? Nothing really follows from that. It doesn't even follow that it is false that $P(n)$ for all $n$: maybe $P(n+1)$ relies on $P(n-1)$ holding.
And showing that both the base and the step don't hold doesn't show any kind of contradiction either. Again, all it shows is that what you're trying to show is false, rather than true.