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If the terms of a sequence $a_n$ converge to $a$, show that $\frac{a_1+a_2+...+a_n}{n}$ also converges to $a$. Also show that the converse isn't necessarily true.

Please help me with this- I am unable to approach this problem, and don't even know where to begin looking for a solution.

Lieutenant Zipp
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V2002
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1 Answers1

1

Hint:

Just remind that if $a_n$ converges to $a$, as of a certain $N$, the terms $a_n$ remain at distance at most $\epsilon$ of $a$. Now if you take the average, you will have a finite number of terms with arbitrary values, but an infinity of them at distance $\epsilon$ or less. So with sufficiently many terms, the average will stay close to $a$.