Provide a convincing argument that if a ϵ R then there is positive integer, n, such that n>a.
My attempt:
If n=a+x where x ϵ R > 0 then n>a for any n>0.
Really unsure though.
As with all mathematics, you must have a list of axioms. We do not have your list.
The completeness axiom states that: Any non-empty set that is bounded above has a least upper bound.
If the statement is false then $a$ is an upper bound for $\mathbb{Z}^+$
So use the axiom and let $k$ be the LUB of $\mathbb{Z}^+$.
Now $k-1<k$ so it is not an upper bound, by the meaning of least.
Thus $\exists j\in\mathbb{Z}^+: k-1<j\le k$.
But that means $k+1<j+1\in\mathbb{Z}^+$
Do you see a contradiction?