If $\{x_n\}$ is convergent and $\{x_ny_n\}$ is divergent, then $\{y_n\}$ is divergent.

Question

If $\{x_n\}$ is convergent and $\{x_ny_n\}$ is divergent, then $\{y_n\}$ is divergent.

Intuitively I believe this is true, since a divergent sequence must have a divergent subsequence. However, I have been reading about cauchy sequences which I dont claim to understand it fully seems to make me doubt my intuitive answer further.

I would appreciate it if someone can shed some light, thank you.

Answer 1

Assume the contrary that $\{y_{n}\}$ is not divergent, then $\{y_{n}\}$ is convergent, then the product $\{x_{n}y_{n}\}$ is convergent since $\{x_{n}\}$ is convergent:

$|x_{n}y_{n}-xy|\leq|x_{n}-x||y_{n}|+|x||y_{n}-y|\leq(\sup_{n}|y_{n}|)|x_{n}-x|+|x||y_{n}-y|$.

Answer 2

Your intuition is correct! You can prove this by contradiction: Suppose that $y_n$ is convergent. Since $x_n$ is also convergent, the sequence $x_n y_n$ converges, a contradiction.

Answer 3

If not then you have two convergent sequences and the product would be convergent. That contradicts your assumption.