There was this question in my analysis exam today. I have a feeling it should be easy but no one I've asked knew how to do it.(adsbygoogle = window.adsbygoogle || []).push({});

We associate to the sequence [itex]\{a_n\}[/itex] the sequence defined by

[tex]b_n=\frac{a_1+a_2+...+a_n}{n}[/tex]

Show that if [itex]\{a_n\}[/itex] converges towards a, then [itex]\{b_n\}[/itex] converges towards a.

I realised that

[tex]b_n=\frac{\sum_{n=1}^{\infty} a_n}{n}[/tex]

or even

[tex]b_n=\frac{\sum_{k=1}^{n} a_k}{\sum_{k=1}^{n} 1}[/tex]

but all my attemps involving epsilon-delta, convergence tests, Cauchy convergence "caracterisation", etc. failed. Please tell me how to do this. Thanks a lot.

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# Question in my test today

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