An implication between two statements concerning integers

Question

Does

"Every non-empty bounded below set of integers has a smallest element"

$\implies$ "If $m,n$ are integers with $m>n$ , then $m-n\ge1$"

? If not then what additional assumption is needed ?

Answer 1

I'm not sure about what is supposed to be known or not in the frame of your question...

But, assume $n<m$ and $m-n<1$ then $m<1+n$ and $n<m<n+1$ which is a contradiction to the fact that $n$ and $m$ are integers...