1. ## Monotone Sequence

Hi. I was wondering if someone could show me how to prove this statement,
let P > 0, prove that (P^(1/n) is a monotone sequence and deduce by using a subsequence that the limit must be 1.

I understand that if 0 < P < 1, then the sequence is increasing and if P > 1, the sequence is decreasing, but how do you show it? Also, how I do use the subsequence to prove its limit?

2. There are many ways to prove the proposition. I suspect that whoever set this for you to do has one particular way in mind. However, here is one way that I like.
Given $\displaystyle p > 1\quad \Rightarrow \quad \ln (p) > 0$.
Then $\displaystyle \begin{array}{rcl} 0 < a < b\quad & \Rightarrow & \quad a\ln (p) < b\ln (p) \\ \quad & \Rightarrow & \quad \ln \left( {p^a } \right) < \ln \left( {p^b } \right) \\ \quad & \Rightarrow & \quad p^a < p^b \\ \end{array}$.

This shows that the sequence $\displaystyle p > 1,\quad \left( {p^{\frac{1}{n}} } \right)$ is decreasing; it is also bounded below by 1. I would use a subsequence converging to 1.

3. What's an appropriate subsequence?

4. I would just use the factor that (say $\displaystyle a\geq 1$) $\displaystyle 1\leq a\leq n$ for sufficiently large $\displaystyle n$ it means $\displaystyle 1\leq a^{1/n} \leq n^{1/n}$ now use the squeeze theorem.

5. Honestly though, what's an easy subsequence of this? We don't know what P is, so I'm not even sure what the terms in this sequence look like. Can you just pick a P and make that a subsequence? Somehow I don't think so.

6. Don't worry about it. I figured something out.