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Let $\{u_n\}$ and $\{v_n\}$ be two sequences satisfying the conditions

  • $\displaystyle\lim_{n\rightarrow\infty}|u_n-v_n|=0$,

  • $\displaystyle\lim_{n\rightarrow\infty}|u_n|=\lim_{n\rightarrow\infty}|v_n|=+\infty$.

Prove that $$ \lim_{n\rightarrow\infty}\left(\frac{u_n}{|u_n|}-\frac{v_n}{|v_n|}\right)=0. $$

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  • $\begingroup$ Are they (purely) real or could they be complex numbers? $\endgroup$ – Aryabhata Mar 21 '12 at 22:22
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Hint: $||u| - |v|| \le |u - v|$, and $$ \frac{u}{|u|} - \frac{v}{|v|} = \frac{u}{|u|} - \frac{v}{|u|} + \frac{v}{|u|} - \frac{v}{|v|}$$ so $$ \left| \frac{u}{|u|} - \frac{v}{|v|}\right| \le \frac{|u - v|}{|u|} + \left| \frac{|v|}{|u|} - 1 \right|$$

The hypotheses could be weakened considerably: it is enough for $|u_n| \to \infty$ with $\frac{|u_n - v_n|}{|u_n|} \to 0$. Moreover, this could be in any normed linear space.

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Hint: What are the possible values of $\frac x{|x|}$?

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    $\begingroup$ That will work for the real case. $\endgroup$ – Robert Israel Mar 21 '12 at 22:23
  • $\begingroup$ @RobertIsrael: good point. I took them as real. $\endgroup$ – Ross Millikan Mar 21 '12 at 22:24

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