$R$ is a relation over $\mathbb Z$.
$aRb \leftrightarrow a - b \le 10$
Verify if it is reflexive, symmetric, transitive, antisymmtetic or total.
I can tell it is reflexive, since $a-a = 0 \le 10$.
It isn't symmetric, since $(0R11)$ but $\lnot(11R0)$
Not antisymmetric either, because $(10R0)\land(0R10)$ but $\lnot(10 = 0)$
Not transitive either, because $(11R1) \land (1R0)$ but $\lnot(11R0)$
All was well, until I tried to verify if it was total. Haven't been able to find a counter-example, so it is likely total after all. However, I don't know how to prove it. How can I?