I am trying to write a disproof for a universal statement I have a statement that says, for all $x$ in the set of integers, there is some $y$ in the set of naturals, such that $y$ is less than $x$.  
I know that this is false because $x$ can be negative, and a positive is not less than a negative.  
I'm not sure how to construct the disproof for this statement. 

You can write it down as follows (then depending from the course you are attending it might or might not be formal enough):
Suppose $x\in\mathbb{Z}$ is negative, and let $y\in\mathbb{N}$. Then
$$x<0\le y\ .$$
Therefore, there is no $y\in\mathbb{N}$ satisfying the statement.
