The reasoning looks ok so far to me, but you have an "if and only if" which means you have to do the reverse. I'd probably try to say what you did a little fancier, like:

Assumemis a proper subset ofn. Then there is some x innthat is not inm(of course we'd use the symbols here, you noe the one that looks like a capital e and then the same symbol with a slash through it). Thus |m|<|n| for allmandn.

Are you sure the question didn't ask |m|<|n|? That seems to be what you're proving.