Disprove that for all $x \in \mathbb{R}_{>0},$ there exists $y \in \mathbb{R}$ such that for all $y<n \in \mathbb{N}, n<x$

Question

So i have been asked to prove this statement in one of my courses.

In class we learned that if we want to disprove for-all statement we should prove there-exist statement, and if we want to disprove there-exist statements we should prove for-all statement.

So to begin my disprove of the statement I started by choosing x = 1/2 and let y be a real number.

But my question is how to use n?

should I prove there-exist statement, because like I said above - when we want to disprove for-all statement we should prove there-exist statement, and by so to choose n to be a specific (natural) number.

or should I let n be a natural number?

I know it may seem a basic question, but I really can't determine why is one of the above correct, and i would like to know in general how to prove this kind of statements.