simple but puzzling proofs

I am a bit puzzled on this. It is a bit embarrassing.

Is the following statement true or false?

For all y there exists an x s.t. x > y^2 + 1

The problem says nothing else about x or y and this is confusing. I have a feeling it is true because I can always find a number that is bigger than another number. Am I thinking correctly? Or is it that I choose y to be zero and x to be zero and thus it is false by counterexample. I think that this is incorrect because it is for some x not for all x.