It is true that the sequence b_n which is the average sequence of the sequence a_n converges to the same limit as the original sequence a_n, that is, lim b_n=lim a_n=a. The proof is slightly involved and can not be written here.
Nikolaos P.
tutor
If you google Cesaro mean you will find plenty of proofs online. In the argument that you wrote above the denominator does go to infinity but the numerator does not have an obvious limit.
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10/13/20
Ashley P.
Could you please provide a link to a source of this proof? Many thanks! What I initially thought was applying limit to the numerator and denominator seperately , which yields lim n->infinity (an)/lim n-> infinity(n) , which is equal to a/infinity->0 Am I missing something here? Could you please explain? Many thanks!10/13/20