|
|
|
|
a prime theorem of a convergent sequence
|
(Theorem)
|
|
Proof. We first show that $(b_n)$ converges to $L$ . Let $\varepsilon>0$ . Select a positive integer $N_0$ such that $n\ge N_0$ implies $|a_n-L|<\varepsilon/2$ . Since $(a_n)$ converges to a finite value, there is a finite $M$ such that $|a_n-L|<M$ for all $n$ . Thus we can select a positive integer $N\ge N_0$ for which $(N_0-1)M/N<\varepsilon/2$ .
By the triangle inequality,
Hence $(b_n)$ converges to $L$ .
To show that $(c_n)$ converges to $L$ , we first define the sequence $(d_n)$ by $d_n=c_n^n=a_1\cdots a_n$ . Since $d_n$ is a positive real sequence, we have that$$ \liminf \frac{d_{n+1}}{d_n} \le \liminf \sqrt[n]{d_n} \le \limsup \sqrt[n]{d_n} \le \limsup \frac{d_{n+1}}{d_n},$$ a proof of which can be found in [1]. But $d_{n+1}/d_n=a_{n+1}$ , which by assumption converges to $L$ . Hence $\sqrt[n]{d_n}=c_n$ must also converge to $L$ . 
- 1
- Rudin, W., Principles of Mathematical Analysis, 3rd ed., McGraw-Hill, New York, 1976.
|
"a prime theorem of a convergent sequence" is owned by georgiosl. [ full author list (3) | owner history (5) ]
|
|
(view preamble | get metadata)
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 3024 times total.
Classification:
| AMS MSC: | 40-00 (Sequences, series, summability :: General reference works ) |
|
|
|
|
|
|
Pending Errata and Addenda
|
|
|
|
|
|
|
|
|
|
|