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Show that there exist unbounded sequences, $x_n\neq y_n$, such that $x_n-y_n\rightarrow 0$ as $n\rightarrow \infty$

How do I prove this? What sort of sequence would even satisfy this, if it is unbounded?

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3 Answers 3

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Work backwards. Choose a sequence $z_n$ that tends to 0 and any unbounded sequence $y_n$. Define $x_n = y_n + z_n$. Then by construction $x_n - y_n = (y_n + z_n) - (y_n) = z_n \to 0$.

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The simplest is perhaps: $$x_n=\sqrt{n+1}, \quad y_n=\sqrt n.$$

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    $\begingroup$ what about $n + \frac1n$ and $n$? :) $\endgroup$
    – Giovanni
    Oct 3, 2015 at 19:21
  • $\begingroup$ Wow, I can't believe none of those occurred to me! Thanks! $\endgroup$
    – Matt G
    Oct 3, 2015 at 19:30
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take $x_n = n + \frac{1}{n}$ and $y_n = n$, then $x_n - y_n = \frac{1}{n}$

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