Supremum of Set of Real Numbers is at least Supremum of Subset/Proof 1

Proof 1
The number $\sup S$ is an upper bound for $S$.

Therefore, $\sup S$ is an upper bound for $T$ as $T$ is a non-empty subset of $S$.

Accordingly, $T$ has a supremum by the Continuum Property.

The number $\sup S$ is an upper bound for $T$.

Therefore, $\sup S$ is greater than or equal to $\sup T$ as $\sup T$ is the least upper bound of $T$.