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Let $x_n$ be a sequence of real strictly positive numbers. If $\lim_{n\to \infty} x_n=\infty$, is it necessary for the sequence to be strictly increasing? If not, give a counterexample. Intuitively I think that it is true, but I have no idea how to prove it.

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  • $\begingroup$ How about $x_n = \begin{cases} n & \text{ if } n \text{ is odd} \\ 0 & \text{ if } n \text{ is even} \end{cases}$? $\endgroup$ – Aniruddha Deshmukh Nov 13 '18 at 15:40
  • $\begingroup$ @Deshmukh Such $x_n$ has not limit $\infty$, maybe $n-1$ for $n$ even would work fine. $\endgroup$ – gimusi Nov 14 '18 at 2:26
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Let consider for example

$$x_n=n+2\cos (n\pi)=-1,4,1,6,3,\ldots$$

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