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Seth|MMORSE
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Homework Statement
Prove the following theorem, originally due to Cauchy. Suppose that [itex](a_{n})[/itex][itex]\rightarrow a[/itex]. Then the sequence [itex](b_{n})[/itex] defined by [itex]b_{n}=\frac{(a_{1}+a_{2}+...+a_{n})}{n}[/itex] is convergent and [itex](b_{n})[/itex][itex]\rightarrow a[/itex].
Homework Equations
A sequence [itex](a_{n})[/itex] has the Cauchy property if, for each [itex]ε>0[/itex] there exists a natural number N such that [itex]|a_{n} - a_{m}|<ε[/itex] for all [itex]n,m>N[/itex]
The Attempt at a Solution
I don't exactly know what am I suppose to do here ... ?
The only thing I could think of is [itex](a_{1}+a_{2}+...+a_{n})≈n*a_{n}[/itex] for a very huge n but that doesn't seem really relevant.
Am I suppose to show that [itex](b_{n})[/itex] is a subsequence? Or [itex](b_{n})[/itex] is somewhat related to the [itex](a_{m})[/itex] in the definition of the Cauchy sequence?
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