Almost, the idea is correct, BUT you're off by a factor of n and the powers of p and q.
The problem is you skipped a step and made 2 errors.
NOW use cheby's or markoff or whatever you want to call it
(b) follows from many results on convergence in probability
OR use the same exact proof. The absolute value kills the minus sign, so there is no distinction...
(c) can be done several ways.
You can just quote that hence
So the difference
Or you can use the rule about products.