My friend and I were talking about math stuff as usual, when he brought up a fake proof for the statement:
There is no real number greater than $0$.
Now obviously this isn't true, because any positive number is $>0$. But I could not argue convincingly enough that his "proof" was wrong, because I can't pinpoint the specific step at which it went wrong.
The following is his "proof":
Suppose that there is at least one real number greater than $0$. Then it follows that if we list out all the real numbers greater than $0$ and sort them in ascending order, then there should be a first number greater than $0$.
Now let's suppose that $\varepsilon$ is the first number in this list. But, $\varepsilon/2$ is smaller than $\varepsilon$ yet still greater than $0$, because $\varepsilon/2$ is $\varepsilon/2$ above $0$, which indicates that $\varepsilon/2$ should be the first number in the list.
This contradicts our original assumption that $\varepsilon$ is the first number, so therefore, there are no real numbers greater than $0$.
I tried to argue that the entire premise of the proof is questionable, because it's impossible to define "first" numbers for infinite lists (like $\Bbb Z$), but he counters by saying that many infinite lists actually do have first numbers, like $\Bbb N$.
What is wrong with my friend's "proof"?