If we are given monotonic real sequences $(a_n)_{n\ge1}, (b_n)_{n\ge1}$ then does the limit $\lim_{n \to \infty} a_n b_n $ always exists (+-infinity is also considered as limit point)? The case that needs looking into is obviously the one when, for example $a_n \to \infty$ and $b_n \to 0$ .
I've been thinking a bit and i can't seem to find a counterexample (i am pretty sure that a limit does not always exist)
Thanks in advance!
Let $a_n=2^{-\lfloor \log_2 n\rfloor}$ and $b_n=n$. Then $\limsup_{n\to\infty} a_nb_n = 2$ and $\liminf_{n\to\infty} a_nb_n=1$.
We see this because $a_nb_n = 2^{\{\log_2 n\}}$ where $\{x\}=x-\lfloor x\rfloor$ is the fractional part of $x$.
If you want strictly monotonic:
$$a_n= \frac{1}{2^{\lfloor\log_2 n\rfloor} n}, b_n=n^2$$
Let us do this when $a_n$ and $b_n$ are positive sequences.
Claim: Let $c_n$ be a positive valued sequence then there are monotonic sequences $a_n,b_n$ such that $c_n=a_n b_n$.
WLOG assume that $a_n$ is decreasing and $b_n$ is increasing. We do this recursively. Assume we have defined the first $k$-terms, then what we need to do is to find $a_{k+1}$ and $b_{k+1}$ such that $c_{k+1}= a_{k+1}b_{k+1}$ and $a_{k+1}$ is the smaller than all $a_1,a_2,\cdots, a_k$ and similarly for $b_{k+1}$. But this is always possible since $c_k=xy$ is a parabola.
Now for your question take $c_n$ non convergent and use the claim.
