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May 22nd, 2017, 08:00 AM  #1 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  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, 09:37 AM  #2 
Math Team Joined: Dec 2013 From: Colombia Posts: 7,042 Thanks: 2344 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 $n$. 
May 22nd, 2017, 09:56 AM  #3 
Senior Member Joined: Nov 2015 From: hyderabad Posts: 206 Thanks: 2  

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