Let $a_1$ be an integer. Then we assume
$$ a_{n+1} = \begin{cases} 3a_n+1,&\text{$a_n$ is odd}\\ \frac{a_n}{2},&\text{$a_n$ is even} \end{cases} $$
Now we prove that
for any $a_1\in\mathbb N$, there exists $N$ which satisfy: $a_n=1,2$ or $4$,$n\geq{N}$.
At first I want to give it a suitable category for the problem: analysis. And I want to use the basic method: evaluate the upper bound for $a_n$, however I find it's not easy because the iteration is rely on the odd or even property of $a_n$. So I attempt the method of number theory. But I failed to find any way to go over it. Can anyone have idea? Thank you.
It's false because if $a_n=1$ then $a_{n+1}=4$.