We discussed problem 5.15 in homework 7. This way of using induction to prove something holds for finitely many objects when you know it holds for two is very common and useful to know.
We also talked about practice problem 3, and discussed the proof of irrationality of the square root of 2. In general, it is not too hard to show the square root of a natural number which is not a perfect square is irrational.
However, it is in general very hard to prove a given real number is irrational. For example, one can (with a lot of work) prove that π and e are irrational, but it is not known whether π + e is.