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Let $a>0$. Be $(a_n)$ a sequence of real numbers . Define $x_n$ as a recurrent sequence : $$x_{n+1}=(1-\dfrac{a}{n})x_n+\dfrac{a_n}{n}$$ Prove that $x_n$ converges to $0$ if and only if sequence $z_n=\dfrac{a_1+a_2+...+a_n}{n}$ converges to $0$.

I know that if we have $t_n$ convergent to a subunitary real (positive or negative) number and if $x_{n+1}-t_nx_n$ is also convergent, then $x_n$ is convergent to a known limit. but here $(1-a/n) \rightarrow 1$ , which is not a subunitary number. Also, the conditions for reverse Stoltz-Cesaro don t seem to be ensured here. any help?

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