You have a little problem here:

You shouldn't assume this. You need to show that this is true, based on the case.now I want to assume this is true for n=k+1

(1/2)+(1/6)+(1/12)+...+1/(k+1)(k+2)=(k+1)/(k+2)

Let's just look at the left hand side and try to work our way to the right.

Look at all of the terms except the last one. By the induction hypothesis, we can substitute them with and receive

Just clean this up and you will get what you want.