Example of two divergent sequences such that their product converges.
I know, if $x_n=\left\{(-1)^n\right\}$ and $y_n=\left\{(-1)^{n+1}\right\}$, then their product converges to $(-1)$. But here $x_n$ and $y_n$ are oscillatory sequences, they are not properly divergent(i.e. they do not diverge to $+\infty$ or $-\infty$.
I want to know, are there two 'properly' divergent sequences so that their product converges? Please anyone help me. Thanks in advance.
The answer to your question is no.
If two sequences diverge to plus or minus infinity the absolute value of products are unbounded.
Since a convergent sequence is necessarily bounded the product of two properly divergent sequences is not convergent.