Supremum of Subset of Real Numbers/Examples/Strictly Negative Real Numbers

Example of Supremum of Subset of Real Numbers

Let $\R_{<0}$ be the (strictly) negative real numbers:

$\R_{<0} := \openint \gets 0$

Then the supremum of $\R_{<0}$ is $0$.

Proof

We have that $0$ is an upper bound of $\R_{<0}$.

Let $x < 0$.

Then $x \in I$.

Then

$\dfrac x 2 > x$

while:

$\dfrac x 2 \in \R_{<0}$

and so $x$ is not an upper bound of $\R_{<0}$.

Hence the result.

Note that the supremum of $\R_{<0}$ is in this case not actually an element of $\R_{<0}$.

$\blacksquare$

Sources

• 1975: W.A. Sutherland: Introduction to Metric and Topological Spaces ... (previous) ... (next): $1$: Review of some real analysis: $\S 1.1$: Real Numbers: Example $1.1.3$