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May 19th, 2017, 09:15 PM   #1
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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
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May 19th, 2017, 10:27 PM   #2
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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.
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May 19th, 2017, 11:17 PM   #3
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Quote:
Originally Posted by Lalitha183 View Post
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

Modified notation with space
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