Definition for supremum Is the below definition for supremum correct? If no, then how to define it in similar way?
$$\sup E = s \Longleftrightarrow(\forall t, \; \forall x, \; x \le t \Longrightarrow x \le s \le t) $$
We suppose that $E \subseteq \mathbb R$; $x\in E$; $s,t \in \mathbb R$.

No. $s$ is $\le$ any other upper bound of $E$. It is not $\le$ any number that happens to be above an element of $E$. For instance if $E = (0,1)$ and $x = 1/2$ then $s = 1$, and $1/2 < 3/4$, but $s \not \le 3/4$.
