Power of a Number Less Than One

Theorem

Let $X$ be one of the standard number fields $\Q, \R, \C$.

Let $x \in X: x > 0$ be a real number such that $\left\vert{x}\right\vert < 1$.

Let $\left \langle {x_n} \right \rangle$ be the sequence in $X$ defined as $x_n = x^n$.

Then $\left \langle {x_n} \right \rangle$ is a null sequence.

Proof

As $\left\vert{x}\right\vert < 1$ it follows that $\left\vert{x}\right\vert = \dfrac 1 y$ where $\left\vert{y}\right\vert > 1$.

So we can express $\left\vert{x}\right\vert$ as $\left\vert{x}\right\vert = \dfrac 1 {1 + h}$ where $y = 1 + h, h > 0$.

(In fact, $h = \dfrac 1 {\left\vert{x}\right\vert} - 1$.)

Then we have:

	\(\displaystyle \)	\(\displaystyle \left({1 + h}\right)^n\)	\(=\)	\(\displaystyle 1 + n h + \frac 1 2 n \left({n-1}\right)h^2 + \ldots + h^n\)	\(\displaystyle \)	Binomial Theorem
	\(\displaystyle \)	\(\displaystyle \)	\(>\)	\(\displaystyle n h\)	\(\displaystyle \)	as $h > 0$

So:

	\(\displaystyle \)	\(\displaystyle \left\vert{x}\right\vert^n\)	\(=\)	\(\displaystyle \frac 1 {\left({1 + h}\right)^n}\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(\le\)	\(\displaystyle \frac 1 {nh}\)	\(\displaystyle \)
	\(\displaystyle \)	\(\displaystyle \)	\(=\)	\(\displaystyle \frac {\left({1 / h}\right)} n\)	\(\displaystyle \)

But $\left \langle {\dfrac 1 n} \right \rangle$ is a null sequence, from the corollary to Power of Reciprocal, so $\dfrac 1 n \to 0$ as $n \to \infty$.

By the Multiple Law it follows that $\dfrac 1 {n h} \to 0$ as $n \to \infty$.

The result follows from the Squeeze Theorem for Sequences.

$\blacksquare$

Notes

This result and Power of Reciprocal are sometimes referred to as the basic null sequences.

Sources

• K.G. Binmore: Mathematical Analysis: A Straightforward Approach (1977)... (previous)... (next): $\S 4.12$