# A product of bounded and null sequences is a null sequence.

Assume $$\{x_k\}$$ and $$\{y_k\}$$ are two sequences in $$\mathbb{R}^n$$ such that the $$\lim_{k \rightarrow \infty}x_k = 0$$ and $$\{y_k\}$$ is bounded. prove $$\lim_{k \rightarrow \infty} (x_k \centerdot y_k) = 0$$

I'm honestly not even sure where to start with this, so we have two convergent sequences but I don't know how to prove the dot product converges to 0 as well

• Try to use $|x_k\cdot y_k|\leq|x_k|\cdot|y_k|$ – Federico Nov 27 '18 at 16:45

We know that $$\forall \epsilon>0\quad \exists M\quad \forall n>M\quad |x_n|<\epsilon$$also $$|y_n| for some $$B>0$$ therefore$$|x_ny_n|=|x_n|\cdot |y_n|=B|x_n|\le B\epsilon$$for $$n>M$$. Therefore$$\lim_{n\to\infty} x_ny_n=0$$