May 19th, 2017, 09:15 PM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  Real Sequence
Let $\displaystyle l \epsilon R $, and $(a_n)$ be a real sequence. Then which of the following condition is equivalent to $\displaystyle '(a_n) \rightarrow l $ as $\displaystyle n \rightarrow \infty' $ A) $\displaystyle \forall \epsilon > 0 , \exists n_0 \epsilon N $ such that $\displaystyle a_n  l < 2\epsilon $ whenever $\displaystyle n \geq n_0. $ B) $\displaystyle \forall \epsilon > 0 , \exists n_0 \epsilon N $ such that $\displaystyle a_n  l < \epsilon $ whenever $\displaystyle n \geq 2n_0. $ C) $\displaystyle \forall \epsilon > 0 , \exists n_0 \epsilon 3N $ such that $\displaystyle a_n  a_m < 2\epsilon $ whenever $\displaystyle m,n \geq n_0. $ D) $\displaystyle \forall \epsilon > 0 , \exists n_0 \epsilon N $ such that $\displaystyle a_n  a_m < 2\epsilon $ whenever $\displaystyle m,n \geq n_0. $ How to check these properties ? Someone help!! Thanks 
May 19th, 2017, 10:27 PM  #2 
Senior Member Joined: Aug 2012 Posts: 1,641 Thanks: 415 
Oh I see, your notation was confusing. You mean $n_0 \in \mathbb N$. Otherwise it just looks like capital $N$; and many books use $N$ to mean what you're calling $n_0$. Also you can use $x \in X$ for set membership, then it puts in the right amount of whitespace. Just some markup hints tonight, no time for the epsilons. Last edited by Maschke; May 19th, 2017 at 10:55 PM. 
May 19th, 2017, 11:17 PM  #3  
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  Quote:
Modified notation with space  

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