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May 22nd, 2017, 07:00 AM   #1
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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.
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May 22nd, 2017, 08:37 AM   #2
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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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May 22nd, 2017, 08:56 AM   #3
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Quote:
Originally Posted by v8archie View Post
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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