order relation and numbers I want to prove the following: If $a$ and $b$ are elements of an ordered field such that $a \leq b+c$ for every $c >0$, then $a \leq b$.
So suppose that $a>b$. Then $a-b > 0$ or $b-a \leq 0$. We know that $a \leq b+c$ so that $b+c-a \geq 0$ or $b-a \geq -c$. How would I get the contradiction?

Suppose $a \gt b$. $ c\gt 0 \implies $ 
$$ a + c \gt b \implies a \gt b-c $$
which is a contradiction.
