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Let $(a_n)$ be a sequence such that $(a_n+a_{n+1})$ is convergent. Prove that $(a_n/n)$ converges to $0$.

$(a_n+a_{n+1})$ convergent to $L$ means that for all $\epsilon$, there exists $N$ such that for all $n\geq N$, $|L-a_n-a_{n+1}|<\epsilon$, or in other words $L-\epsilon<a_n+a_{n+1}<L+\epsilon$. How to proceed from here?

D.R.
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Paul S.
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2 Answers2

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Hint: $(a_n+a_{n+1})\to 0$ implies $(a_{n+2}-a_n)\to 0$, so there exists $N$ such that, for any $n>N$, $|a_{n+2}-a_n|<\epsilon$. Note that $a_n=(a_n-a_{n-2})+(a_{n-2}-a_{n-4})+\ldots+(a_{N+2}-a_N)+a_N$(here I assume n and N are both even or odd, in the other case we can do this similarly), then try to prove by definition(using $\epsilon-N$ language).

Coiacy
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After you established $a_{n+2}-a_n\to 0,$ one can use Stoltz-Cesaro theorem for even and odd numbers separately: $$\lim_{k\to \infty}\frac{a_{2k}}{2k}=\lim_{k\to \infty}\frac{a_{2k+2}-a_{2k}}{2}=0.$$

leshik
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