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I am trying to solve the following problem in real analysis:

Let $(a_{n})$ be a non-negative sequence in which $a_{n+m} \leq a_{n}a_{m} \; \forall n,m \in \mathbb{N}$.
Show that the sequence $(\sqrt[n]{a_{n}})$ converges and that $\displaystyle \lim_{n \to \infty}\sqrt[n]{a_{n}}=\inf\{\sqrt[n]{a_{n}}|n \in \mathbb{N}\}$.

So far, I have deduced from the assumption $\displaystyle a_{n+m} \leq a_{n}a_{m} \; \forall n,m \in \mathbb{N}$ that $\displaystyle a_{n} \leq a_{1}^{n} \; \forall n \in \mathbb{N}$. So $(a_{n})$ grows at most exponentially. Also, since $\displaystyle \sqrt[n]{a_{n}}=e^{\frac{\log(a_{n})}{n}}$, I'm trying to show that the sequence $\displaystyle \frac{\log(a_{n})}{n}$ either converges or diverges to $-\infty$ instead, but I haven't been able to do so.

Any help would be greatly appreciated. Thank you.

V.G
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A0710046
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1 Answers1

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I introduce you to the fantastic Fekete's lemma here.

First, suppose that $a_N = 0$ for some $N$. By the condition, $a_{N+1} \leq a_Na_1 = 0$ so $a_{N+1} = 0$. By induction, the sequence is $0$ after this point, so of course $\lim_{n} \sqrt[n]{a_n} = 0 = \inf\left\{\sqrt[n]a_n : n \in \Bbb N\right\}$.

Otherwise, all the $a_i$ are positive , so the quantities $\ln a_n$ are well defined.The condition $a_{n+m} \leq a_na_m$ implies that $\log a_{n+m} \leq \log a_n + \log a_m$. Thus, the function $f(n) = \log a_n$ is a sub-additive function.

By Fekete's lemma, $\lim_{n\to \infty} \frac{f(n)}n = \inf_{n} \frac{f(n)}{n}$, translating to $\lim_{n \to \infty} \log (a_{n})^{\frac 1{n}} = \inf_{n} \log (a_n)^{\frac 1n}$.

Now, $\lim_{n \to \infty} (a_n)^{\frac 1n}$ exists and equals $\exp(\lim_{n \to \infty} \log (a_n)^{\frac 1n})$ by continuity. By the condition we derived, this equals $\exp(\inf_n \log (a_n)^{\frac 1n})$, but $\exp$ is an increasing function, so this quantity equals $\inf_n (a_n)^{\frac 1n}$.

On Fekete's lemma, it is often used in graph theory to show bounds on certain quantities pertaining to graphs (like escape probabilities in infinite graphs, and so on). It has been found that even in difficult models the lemma actually gives excellent bounds. It is also used in combinatorics and Ramsey theory.

Sarvesh Ravichandran Iyer
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