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There are 3 sequences $a_n, $, $b_n$, $c_n$ with $n \in \mathbb{N} \setminus \{0\}$ given by recursions:

  • $a_{n+1} = \frac{a_n+b_n+c_n}{3}$
  • $b_{n+1} = \sqrt[3]{a_n*b_n*c_n}$
  • $c_{n+1} = \frac{3}{\frac{1}{a_n} + \frac{1}{b_n}+\frac{1}{c_n}}$

Elements $a_1, b_1$ and $c_1$ are all positive numbers. Prove that at least $2$ of those $3$ sequences are convergent and that: $$\lim_{n \to +\infty} a_n = \lim_{n \to +\infty} b_n = \lim_{n \to +\infty} c_n $$

The first thing that I see is that those seqences are made of subsequent elements of functions known as arithmetic, geometric and harmonic means. Because of inequality of means, I know that: $$a_{n+1} = \frac{a_n+b_n+c_n}{3} \geq b_{n+1} = \sqrt[3]{a_n*b_n*c_n} \geq c_{n+1} = \frac{3}{\frac{1}{a_n} + \frac{1}{b_n}+\frac{1}{c_n}}$$

Then, I know for sure that:

  • every subsequent element of $a_n$ must be smaller than the former one because $b_n$ and $c_n$ are smaller than $a_n$ for every n
  • every subsequent element of $c_n$ must be greater than the former one because $a_n$ and $b_n$ are bigger than $c_n$ for every n

Additionally, I know that the limit of each one of those means is reached when all 3 elements in the mean are equal. In my case all 3 elements are the same for all 3 means so, the limit must be equal as well.

But here comes my problem - how to write it all down, to present my thinking in a correct way? Is there any scheme that I that could apply in my case (I only know how to formally find limits and prove convergence in cases of $\lim_{n \to +\infty}$ for sequences of $n$ elements)?

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If you put $a_n \gt$ in front of your inequalities and $\gt c_n$ afterward you have a proof of convergence for $a$ and $c$ as the sequences are monotone and bounded.

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  • $\begingroup$ And do I need to show in any way that the sequences are monotone and bounded? I mean, it is rather obvious in that case (they are means of means) but who knows. $\endgroup$
    – theboyboy
    Commented Nov 22, 2020 at 16:48
  • $\begingroup$ You said that $a_{n+1} \lt a_n$, which says the $a$ sequence is monotone decreasing and that $c$ is monotone increasing. $\endgroup$ Commented Nov 22, 2020 at 16:51

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