Supremum of Singleton

Theorem
Let $\left({S, \preceq}\right)$ be an ordered set.

Then for all $a \in S$:


 * $\sup \left\{{a}\right\} = a$

where $\sup$ denotes supremum.

Proof
Since $a \preceq a$, $a$ is an upper bound for $\left\{{a}\right\}$.

Let $b$ be another upper bound for $\left\{{a}\right\}$.

Then necessarily $a \preceq b$.

It follows that indeed:


 * $\sup \left\{{a}\right\} = a$

as desired.

Also see

 * Infimum of Singleton