Supremum of Subset of Real Numbers/Examples/Example 5

Example of Supremum of Subset of Real Numbers
The subset $S$ of the real numbers $\R$ defined as:
 * $S = \set {x \in \R: x^2 \le 2 x - 1}$

admits a supremum:


 * $\sup S = 1$

such that $\sup S \in S$.

Proof
But as $\forall x \in \R: \paren {x - 1}^2 \ge 0$ it follows that:


 * $\paren {x - 1}^2 \le 0 \implies x - 1 = 0$

and so:
 * $\set {x \in \R: x^2 \le 2 x - 1} = \set 1$

and so trivially:


 * $\sup \set {x \in \R: x^2 \le 2 x - 1} = 1$

and as $1 \in \set 1$ it follows that $\sup S \in S$.