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 May 22nd, 2017, 07:00 AM #1 Senior Member   Joined: Nov 2015 From: hyderabad Posts: 242 Thanks: 4 Relation of two sequences Let ${x_n}$ and ${y_n}$ be two sequences in $R$ such that $\displaystyle \lim_{n \rightarrow \infty} x_n = 2$ and $\displaystyle \lim_{n \rightarrow \infty} y_n = -2$. Then A) $\displaystyle {x_n} \geq {y_n}$ for all $\displaystyle n \in N$ B) $\displaystyle {x_n}^2 \geq {y_n}$ for all $\displaystyle n \in N$ C) there exists an $\displaystyle m \in N$ such that $\displaystyle |x_n| \leq {y_n}^2$ for all $n>m.$ D) there exists an $\displaystyle m \in N$ such that $\displaystyle |x_n| = |y_n|$ for all $n>m.$ My answer : Option D Kindly check and let me know if I'm wrong. Thanks
 May 22nd, 2017, 08:37 AM #2 Math Team   Joined: Dec 2013 From: Colombia Posts: 7,681 Thanks: 2659 Math Focus: Mainly analysis and algebra No. C. There is no need for sequences that tend to a common limit to share elements. e.g. $\displaystyle \lim_{n \to \infty} c = \lim_{n \to \infty} (c-\tfrac1n)$ but $c-\tfrac1n \ne c$ for all $n$. Thanks from Lalitha183
May 22nd, 2017, 08:56 AM   #3
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 Originally Posted by v8archie No. C. There is no need for sequences that tend to a common limit to share elements. e.g. $\displaystyle \lim_{n \to \infty} c = \lim_{n \to \infty} (c-\tfrac1n)$ but $c-\tfrac1n \ne c$ for all $n$.
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