True or False: If $\lim_{n\to\infty}a_n=\infty$ then exists $M>0$ such that $a_{n+1}>Ma_n$ for almost all $n$?

Question

True or False: If $\lim_{n\to\infty}a_n=\infty$ then exists $M>0$ such that $a_{n+1}>Ma_n$ for almost all $n$?

Answer 1

Try $$ a_n=\begin{cases}n^2&n\text{ even}\\n&n\text{ odd}\end{cases}$$

Answer 2

What if you define $(a_n)_{n\in\mathbb N}$ as $a_n=2n+n(-1)^n$?

Answer 3

Take $$u_{2n} = \ln n~~~u_{2n+1} = e^n$$