Does $\lim a_n$ exist?

Question

Let $(a_n)_n$ and $(b_n)_n$ be two positive real sequences.

If $0\leq a_n \leq b_n$ and $\lim b_n$ exists. Is $\lim a_n$ exists?

I ask this question because now I'm reading a paper and I think the authors used the above properties, but I think it's not in general true.

Answer 1

Of course it is not true in general. Take $a_n=1+(-1)^n$ and $b_n=2$. Then your conditions are satisfied, but $\lim_{n\to\infty}a_n$ does not exist.

Answer 2

The only thing that you can say is probably that $\limsup\limits_{n\rightarrow\infty}a_n\leq \lim\limits_{n\rightarrow\infty} b_n$.