July 3rd, 2017, 08:25 AM  #1 
Senior Member Joined: Dec 2015 From: Earth Posts: 177 Thanks: 23  Limit consequence
Is consequence true ? If $\displaystyle \lim_{n\rightarrow \infty} \frac{a_n}{b_n}=0 \; \; \Rightarrow \; \; b_n\geq a_n$ 
July 3rd, 2017, 09:17 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,042 Thanks: 2344 Math Focus: Mainly analysis and algebra 
No. $a_n = 0, b_n = 1$ If you meant to have both positive \begin{align*} a_n &= n \\ b_n &= \begin{cases} n^2 & (n \gt N) \\ 1 & (n \le N) \end{cases} \end{align*} The answer to the question you intended to ask (i.e. for sufficiently large $n$) is yes. Think about $\delta\epsilon$ definition. $\displaystyle \lim_{n \to \infty} \frac{a_n}{b_n} = 0$ means that for every $\epsilon \gt 0$ there exists an $N \in \mathbb N$ such that $\frac{a_n}{b_n} \lt \epsilon$ for all $n \gt N$. Pick $ \epsilon = 1$ and you have your result. Last edited by v8archie; July 3rd, 2017 at 09:40 AM. 
July 3rd, 2017, 09:39 AM  #3 
Senior Member Joined: Dec 2015 From: Earth Posts: 177 Thanks: 23 
i forgot to check $\displaystyle a_n,b_n >0$

July 3rd, 2017, 09:40 AM  #4 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,042 Thanks: 2344 Math Focus: Mainly analysis and algebra 
See update.

July 4th, 2017, 07:40 AM  #5  
Senior Member Joined: Mar 2015 From: New Jersey Posts: 1,175 Thanks: 90  Quote:
$\displaystyle \left  \frac{a_{n}}{b_{n}} \right \leq \epsilon\\ a_{n} \leqb_n$, n>N Ref: v8archie post#2. Last edited by zylo; July 4th, 2017 at 07:51 AM. Reason: add n>N  
July 24th, 2017, 05:11 AM  #6 
Math Team Joined: Jan 2015 From: Alabama Posts: 2,829 Thanks: 753 
Knowing the limit of a sequence does not tell you anything about all of the numbers in the sequence you can change any finite number of terms in the sequence arbitrarily without changing the limit.


Tags 
consequence, limit 
Search tags for this page 
Click on a term to search for related topics.

Thread Tools  
Display Modes  

Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
upper limit = lower limit implies convergence  zylo  Calculus  13  May 31st, 2017 01:53 PM 
Limit help!!!  jasmin99  Calculus  2  March 30th, 2014 08:05 PM 
Limit Superior and Limit Inferior  veronicak5678  Real Analysis  4  August 22nd, 2011 11:07 AM 
limit  panky  Calculus  9  July 22nd, 2011 05:11 PM 
when should we evaluate left limit and right limit?  conjecture  Calculus  1  July 24th, 2008 02:14 PM 