Let $(a_n)$ be a monotone and bounded sequence such that $a_n \to a$. Let $(b_n)$ be defined as $b_n = (a_1 + a_2 + ... + a_n)/n$. I know $(b_n)$ is monotone and bounded, but how do I prove that $b_n \to a$ using the definition of a Cauchy sequence?
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Welcome to Math Stack Exchange. Your post would look nicer if you used MathJax – J. W. Tanner Feb 08 '19 at 01:51
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1Just some general remarks: 1.) If $(a_n)$ converges, it is redundant to demand that it is bounded. 2.) You can show that $(b_n)$ is a Cauchy sequence in order to prove that it converges (assuming you are working in $\mathbb R$), but this way you will not directly be able to say anything about its limit. – Mars Plastic Feb 08 '19 at 01:55
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Possible duplicate of arithmetic mean of a sequence converges – YuiTo Cheng Feb 08 '19 at 02:39
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You cannot prove that $b_n \to a$ using definition of a Cauchy sequence. Cauchy property can be used to prove convergence but it does not identify the limit. – Kavi Rama Murthy Feb 08 '19 at 05:53