Supremum of Subset of Real Numbers/Examples/Example 8

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

does not admit a supremum.

Proof
Let $x = n \pi$.

From Sine of Multiple of Pi, we have that:
 * $\sin x = 0$

and so:
 * $x \sin x = 0$

That is:
 * $\forall n \in \Z: n \pi \sin n \pi < 1$

and $n \pi$ is not bounded above.