Supremum of Subset of Real Numbers/Examples/Example 3

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Example of Supremum of Subset of Real Numbers

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

$V := \set {x \in \R: x > 0}$

does not admit a supremum.


Proof

Aiming for a contradiction, suppose $x \in \R$ is a supremum for $V$.

Then we have that:

$x + 1 \in V$

and it is seen that $x$ is not a supremum after all.

$\blacksquare$


Sources