# Sequence with limit $\infty$

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.

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

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