Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

Let $a>0$, and let $x$ be a real number. Prove that if $\{r_{n} \}$ is any decreasing rational sequence with limit $x$, then $a^x = \lim_{n \rightarrow \infty} a^{r_n}$

Where in the book $a^x$ is defined as $\lim_{n \rightarrow \infty} a^{s_n}$ where $ \{ s_n \}$ is an increasing sequence.

share|cite|improve this question
up vote 1 down vote accepted

Note that $\{-r_n\}$ is an increasing sequence of rationals that converges to $-x$

Using the definition of your book, we know that: $$lim_{n\rightarrow\infty}a^{-r_n}=a^{-x}>0$$

Since the limit is positive, thus: $$lim_{n\rightarrow\infty}a^{r_n}=a^{x}$$

share|cite|improve this answer
I see how you got $lim_{n \rightarrow \infty} a ^{\r{n}} = a^{-x} > 0$ but I don't know why it implies the second part thanks. – Jmaff Dec 26 '12 at 22:58
Take the multiplicative inverse of both sides – Amr Dec 26 '12 at 23:21
hmm, still unsure. If I take the inverse of the first line then we get $(lim_{n \rightarrow \infty} a^{r_{n}}) \cdot a^{x} =1$ Or should I look at the inverses of the actual sequence terms? – Jmaff Dec 26 '12 at 23:34
The inverses of the actual terms of the sequence (you can take their inverse becasue they are non zero) – Amr Dec 26 '12 at 23:35
Okay,so since we know that the $\{ -r_n \}$ sequence has an invertible limit, and that each element of the sequence is invertible we know that the limit of the sequence of the inverses has the limit that is the inverse of$\{ -r_n \}$'s limit ? Is this a commonly proved statement? I think it can be prove using the limit rule for limits maybe,Thanks – Jmaff Dec 26 '12 at 23:43

Since $2x-r_n$ is increasing and

$$\lim_n 2x-r_n=x$$

we have

$$a^x=\lim_n a^{2x-r_n}=\lim_n \frac{a^{2x}}{a^{r_n}}= \frac{a^{2x}}{\lim_n a^{r_n}} \,.$$

Multiply both sides by $\frac{\lim_n a^{r_n} }{a^x}$

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.