Supremum of Subset of Real Numbers/Examples/Left Half Open Interval from 0 to 1

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 $\hointl 0 1$.