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 $0$.
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 $1.1.3$