For all $y$, there exists an $x$ where $x\geq y$

Question

For all $y$, there exists an $x$ where $x\geq y$

Is this statement true or false? If so why?

My note says it's true, but I don't really get why. Thanks!

Answer 1

The setting needs to be specified. For example, we could be working with natural numbers, and $x\ge y$ might mean the ordinary $\ge$ relationship.

The sentence says that given any specific $y$, we can produce an $x$ such that $x\ge y$.

Imagine you are given the number $17$. Can you produce an $x$ such that $x\ge 17$? Sure, easy, pick $x=17$. There are many other choices for $x$ possible, such as $x=999$. We opted for the simplest one.

Given any other value of $y$, we could produce an $x$ such that $x\ge y$: just pick $x=y$.

Answer 2

If you choose any number $y$, you can always choose a larger number, as there are infinitely many numbers, or you can always let $x$ equal the number $y$ you chose.