suppose $(a_n)_{n\in\mathbb{N}}$ is a positive sequence and tends to $+\infty$ as $n\to +\infty$. I am wondering about the behaviour of the quotient $\dfrac{a_n}{a_{n+1}}$. Intuitively, I would say that it converges, because both numerator and denominator grow "with the same speed" so for large $n$ we have $\dfrac{a_n}{a_{n+1}}\in \mathcal{O}(1)$. Rigorously, I tried to approach the question by the fact that after some $N\in\mathbb{N}$ the sequence $(a_n)_{n\geq N}$ is monotonone. But I didn't get any further. Could you help me here?
Is the claim even correct?
Let $a_1=1, a_{n+1}=2a_n$ for $n$ even and $a_{n+1}=3a_n$ for $n$ odd. This is a counterexample, right?