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

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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 $0$.


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

Let $x < 0$.

Then $x \in I$.


$\dfrac x 2 > x$


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

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 $\R_{<0}$.