First, check that the start and end of your proof look something like the following:
Proof: Let (a,b) and (c,d) be elements of \(\mathbb{Z}^2\). Suppose that \((a,b) \ll (c,d)\) and \((c,d) \ll (a,b)\).
.....
So (a,b)=(c,d).
If that's not true, you have probably based your proof outline on this version of the definition of antisymmetric:
For any x and y in A, if \(x\not = y\) and \(xRy\), then \(y\not R x\).
Don't use this version of the definition for writing proofs. It will frequently get you into situations that you won't be able to write your way out of. Even when someone makes it work, their proof is usually much harder to read than it should be. Go back and rewrite your proof using this version of the definition:
For any x and y in A, if \(xRy\) and \(yRx\), then \(x=y\).
Once you've fixed this, look at the annotated solutions