Power of Real Number between Zero and One is Bounded

Theorem
Let $x \in \R$ be a real number.

Let $0 < x < 1$.

Let set $S = \left\{{x^n: n \in \N}\right\}$.

Then:
 * $\inf S = 0$

and:
 * $\sup S = 1$

where $\inf S$ and $\sup S$ are the infimum and supremum of $S$ respectively.

Proof
When $n = 0$ it follows that $x^n = 1$ and so $\sup S \ge 1$.

Let $x^k \in S$.

As $x < 1$, it follows from Real Number Ordering is Compatible with Multiplication that:


 * $x^{k+1} < x^k$

So:
 * $\forall x \in S: x \le 1$

Hence it follows that $\sup S = 1$.

As $x > 0$, it follows by Real Number Ordering is Compatible with Multiplication that:
 * $\forall x \in S: x \ge 0$

Therefore $x$ is a lower bound of $S$.

Now suppose $h > 0$ is also a lower bound of $S$.

Then:
 * $\forall n \in \N: x^n \ge h$

and:
 * $\displaystyle \forall n \in \N: \left({\frac 1 x}\right)^n \le \frac 1 h$

But as $0 < x < 1$ it follows that:
 * $\dfrac 1 x > 1$

Thus $\dfrac 1 h$ is an upper bound for $\displaystyle \left\{{\left({\frac 1 x}\right)^n: n \in \N}\right\}$.

From Power of Real Number greater than One is Unbounded Above, this has been shown to be unbounded above.

Therefore there can be no such lower bound $h > 0$ of $S$.

Hence $\inf S = 0$.