2
$\begingroup$

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?

$\endgroup$
7
$\begingroup$

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 $x_n - y_n = (y_n + z_n) - (y_n) = z_n \to 0$.

$\endgroup$
1
$\begingroup$

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

$\endgroup$
  • 6
    $\begingroup$ what about $n + \frac1n$ and $n$? :) $\endgroup$ – Giovanni Oct 3 '15 at 19:21
  • $\begingroup$ Wow, I can't believe none of those occurred to me! Thanks! $\endgroup$ – Matt G Oct 3 '15 at 19:30
0
$\begingroup$

take $x_n = n + \frac{1}{n}$ and $y_n = n$, then $x_n - y_n = \frac{1}{n}$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.