some things you need to add, for clarity:

since a,b are assumed distinct, without loss of generality you may assume a < b (or else switch them, a standard tactic).

you should show why p√2/q is not rational (it's not hard, and only takes a line or two).

p needs to be non-zero, or else your argument fails. for example, what if a = -1/n, and b = 1/n, where n is a VERY large positive integer (like 3 billion)? this is a rather serious defect.

what can you do about this?