Let $a_j$ be a sequence of real numbers. Define $$m_j=\frac{a_1+a_2+\dots +a_j}{j}$$. Prove that, if $\lim_{j\rightarrow\infty}a_j= \ell$, then $\lim_{j\rightarrow\infty}m_j= \ell$. Give an example to show that the converse is not true.

I have been thinking about this question a lot. I know I could rewrite this as $$m_j=\frac{\sum_{j=1}^\infty a_j}{j}$$ But I am having a really hard time after that. Thanks for any help you can give.


marked as duplicate by anomaly, Stefan4024, Arnaud D., Did real-analysis Sep 19 '16 at 16:45

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    $\begingroup$ What have you tried, beyond just (incorrectly) changing the notation in the definition of $m_j$? You clearly need the fact that $\lim_{j\to\infty} a_j = \ell$; what have you done with it? $\endgroup$ – anomaly Sep 19 '16 at 16:33
  • $\begingroup$ I could take the limit of both sides I suppose to get that $\lim m_j=\frac{l}{\lim j}$ $\endgroup$ – MathIsHard Sep 19 '16 at 16:34
  • $\begingroup$ That doesn't make any sense. The limit of $a_1 + \cdots + a_j$ as $j\to\infty$ is not $\ell$, and $\lim_{j\to\infty} j$ is infinite. $\endgroup$ – anomaly Sep 19 '16 at 16:36
  • $\begingroup$ Thanks so much. I search for this before posting and couldn't find those. I am sorry to take your time... $\endgroup$ – MathIsHard Sep 19 '16 at 16:40