Let $E$ be a non-empty subset of an ordered space. Suppose that a is a lower bound of $E$ and $\beta$ is a upper bound of $E$. Show that $ \alpha \leq \beta $.
(1) If $\alpha$ is a lower bound of $E$ then $\forall x\in E\quad x\geq \alpha$
(2) If $\beta$ is a upper bound of $E$ then $\forall x\in E\quad x\leq \beta$
From (1), (2) and since $E$ is ordered then
$$\alpha\leq x\leq\beta\Rightarrow \alpha\leq\beta$$
I'm starting now with real analysis and I'm still learning the art of demonstrating. Is this right? That's enough?