Greatest Element is Supremum/Examples

Examples of Use of Greatest Element is Supremum

Arbitrary Example $1$

Let $S$ be the subset of the real numbers $\R$ defined as:

$S = \set {1, 2, 3}$

Then the greatest element of $S$ is $3$.

From Greatest Element is Supremum it follows that:

$\sup S = 3$


Arbitrary Example $2$

Let $V$ be the subset of the real numbers $\R$ defined as:

$V := \set {x \in \R: x > 0}$

From Supremum of Subset of Real Numbers: Example 3, $V$ has no supremum.

It follows from Greatest Element is Supremum that $V$ has no greatest element either.

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