As mentionned in the geometric sequence entry,

(1)

We first remark, that for the values $s > 1$ we have $\displaystyle\lim_{n\to\infty}s^n = \infty$ (cf. limit of real number sequence). In fact, if $M$ is an arbitrary positive number, the binomial theorem (or Bernoulli's inequality) implies that $$s^n = (1+s-1)^n > 1^n+\binom{n}{1}(s-1) = 1+n(s-1) > n(s-1) > M$$ as soon as $\displaystyle n > \frac{M}{s-1}$ .

Let now $|r| < 1$ and $\varepsilon$ be an arbitrarily small positive number. Then $\displaystyle|r| = \frac{1}{s}$ with$s > 1$ . By the above remark, $$|r^n| = |r|^n = \frac{1}{s^n} < \frac{1}{n(s-1)} < \varepsilon$$ when $\displaystyle n > \frac{1}{(s-1)\varepsilon}$ . Hence, $$\lim_{n\to\infty}r^n =0,$$ which easily implies (1) for any real number $a$ .