May 18th, 2018, 10:29 AM  #1 
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,399 Thanks: 100  Limits Observation
Let {$\displaystyle a_{n}$} be a sequence. $\displaystyle \lim_{n\rightarrow \infty}a_{n}=L$ if, given $\displaystyle \epsilon$, N exists st $\displaystyle a_{n}L < \epsilon$ for all n > N but there are an infinite number of sequences which satisfy this condition, NO MATTER WHAT $\displaystyle \epsilon$ is, namely all sequences which are the same up to $\displaystyle a_{N}$, so you can never arrive at a unique sequence for L. This has implications for decimal representation of numbers. I haven't thought through implications for Cauchy sequences and completeness. Offhand, it looks like the only argument for completeness is Dedekind cuts. 
May 18th, 2018, 11:42 AM  #2  
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  Quote:
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May 18th, 2018, 11:54 AM  #3 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,342 Thanks: 2465 Math Focus: Mainly analysis and algebra 
There has never been any suggestion that there was a unique sequence for each $L$. That's trivial. $$\left.\begin{aligned}\lim_{n \to \infty} a_n &= L \\ \lim_{n \to \infty} b_n &= 0 \end{aligned}\right\} \implies \lim_{n \to \infty} (a_n +b_n) = \lim_{n \to \infty} (a_n b_n) = L$$


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