I was given that $(F,\leq)$ is an ordered field. I was able to prove the following statement:$$\forall a\in F:[\forall x \in F: x>0 \rightarrow x\geq a] \leftrightarrow a\leq 0$$
I was asked, whether this also applies to totally ordered sets or not, but I cannot really find, how to verify that. Isn't the only difference that if I use totally ordered sets, I can't just simply (for example) add a number to both sides of the inequality, which I however can do in an ordered field?
HINT: For a counterexample try the linear order $\Bbb Z$. If that’s not enough of a hint, I’ve left a bit more in the spoiler-protected block below.