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

Note that

And it is shown.