Supremum of Subset of Real Numbers/Examples/Left Half Open Interval from 0 to 1
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Example of Supremum of Subset of Real Numbers
Let $\hointl 0 1$ denote the left half-open real interval:
- $\hointl 0 1 := \set {x \in \R: 0 < x \le 1}$
Then the supremum of $\R_{<0}$ is $1$.
Proof
We have that $1$ is an upper bound of $\hointl 0 1$.
Let $0 < x \le 1$.
Then $x \in I$.
Then:
- $x < \dfrac {1 + x} 2 < 1$
and so $x$ is not an upper bound of $\R_{<0}$.
Hence the result.
Here we see that the supremum of $\hointl 0 1$ is in this case an element of $\hointl 0 1$.
$\blacksquare$
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 $1.1.3$