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 May 19th, 2017, 08:15 PM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 230 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, 09:27 PM #2 Senior Member   Joined: Aug 2012 Posts: 1,858 Thanks: 513 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 09:55 PM.
May 19th, 2017, 10:17 PM   #3
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 Originally Posted by Lalitha183 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) $\displaystyle \forall \epsilon > 0 , \exists n_0 \in N$ such that $\displaystyle |a_n - l| < 2\epsilon$ whenever $\displaystyle n \geq n_0.$ B) $\displaystyle \forall \epsilon > 0 , \exists n_0 \in N$ such that $\displaystyle |a_n - l| < \epsilon$ whenever $\displaystyle n \geq 2n_0.$ C) $\displaystyle \forall \epsilon > 0 , \exists n_0 \in 3N$ such that $\displaystyle |a_n - a_m| < 2\epsilon$ whenever $\displaystyle m,n \geq n_0.$ D) $\displaystyle \forall \epsilon > 0 , \exists n_0 \in N$ such that $\displaystyle |a_n - a_m| < 2\epsilon$ whenever $\displaystyle m,n \geq n_0.$ How to check these properties ? Someone help!! Thanks

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