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Given $(s_n)_{n \geq 1}$, a sequence of positive integers and the inequality $s_ns_m \leq s_{n+m}$ for all integers $n, m$, how does one go about proving that the sequence $(\sqrt[n]{s_n})_{n \geq 1}$ converges?

It is quite easy to see that the sequence is monotone increasing, so all we need is an upper bound for all of its terms. I don't quite see how to do this, since the inequality only seems to give a lower bound..

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Consider the sequence $s_n = 2^{n^2}$ and observe that \begin{align} s_ns_m = 2^{n^2+m^2} \leq 2^{(m+n)^2}= s_{n+m} \end{align} but we see that \begin{align} \sqrt[n]{s_n} = 2^n\rightarrow \infty. \end{align}

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  • $\begingroup$ Thank you for the counterexample. There must have been a mistake in the book I'm reading. $\endgroup$ – Kevin Hsu May 19 '18 at 9:23

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