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I'm trying to solve this question and I'm stuck for whatever reason, I'm not really sure how to start facing this problem, so here's what I've been thinking, given that $b_n$ converges, we know that it's bounded, so there exists some number $k$, so that $-k<b_n<k$, plugging in $b_n$ we get that $-nk<a_1+a_2+...+a_n<nk$. I can't really wrap my head about where I need to go from here (not sure I'm even headed in the right direction), would love some feedback/tips. Thanks

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    $\begingroup$ Use $\lim_{n \to \infty} (b_n - b_{n - 1}) = 0$, then manipulate the difference to try and show what you want. $\endgroup$ – Pratyush Sarkar Nov 27 '19 at 19:24
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Try this: $$b_n = \frac{n - 1}{n} b_{n - 1} + \frac{a_n}{n} $$

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  • $\begingroup$ Thanks! Managed to get to that and prove it successfully! $\endgroup$ – user1780131 Nov 28 '19 at 16:57

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