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a prime theorem of a convergent sequence
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(Theorem)
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Proof. We first show that  converges to  . Let
 . Select a positive integer  such that implies
 . Since  converges to a finite value, there is a finite  such that  for all  . Thus we can select a positive integer  for which
 .
By the triangle inequality,
Hence  converges to  .
To show that converges to , we first define the sequence by
. Since is a positive real sequence, we have that
a proof of which can be found in [ 1]. But
 , which by assumption converges to  . Hence
![$ \sqrt[n]{d_n}=c_n$ $ \sqrt[n]{d_n}=c_n$](http://images.planetmath.org:8080/cache/objects/6494/l2h/img33.png) must also converge to  . 
- 1
- Rudin, W., Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976.
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"a prime theorem of a convergent sequence" is owned by georgiosl. [ full author list (3) | owner history (5) ]
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(view preamble)
Cross-references: proof, triangle inequality, finite, implies, integer, geometric means, arithmetic means, converges, sequence, real, positive
This is version 21 of a prime theorem of a convergent sequence, born on 2004-11-18, modified 2006-11-14.
Object id is 6494, canonical name is APrimeTheoremOfAConvergentSequence.
Accessed 2642 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
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Pending Errata and Addenda
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