Let me illustrate in a lazy way.
What we are being asked to do is choose 2 out of n. C(n,2).
As you can see, everything cancels in the numerator and denominator except
Check it, how many 2 subsets can be made from 10. C(10,2)=45
Now, we want to show an induction step.
See what cancels this time?. Everything but
And it is shown.