# Prove that, if $\lim_{j\rightarrow\infty}a_j= \ell$, then $\lim_{j\rightarrow\infty}m_j= \ell$. [duplicate]

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 StackExchange.ready(function() { if (StackExchange.options.isMobile) return; $('.dupe-hammer-message-hover:not(.hover-bound)').each(function() { var$hover = $(this).addClass('hover-bound'),$msg = $hover.siblings('.dupe-hammer-message');$hover.hover( function() { $hover.showInfoMessage('', { messageElement:$msg.clone().show(), transient: false, position: { my: 'bottom left', at: 'top center', offsetTop: -7 }, dismissable: false, relativeToBody: true }); }, function() { StackExchange.helpers.removeMessages(); } ); }); }); Sep 19 '16 at 16:45

• 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? – anomaly Sep 19 '16 at 16:33
• I could take the limit of both sides I suppose to get that $\lim m_j=\frac{l}{\lim j}$ – MathIsHard Sep 19 '16 at 16:34
• 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. – anomaly Sep 19 '16 at 16:36