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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

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    $\begingroup$ Try to use $|x_k\cdot y_k|\leq|x_k|\cdot|y_k|$ $\endgroup$ – Federico Nov 27 '18 at 16:45
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We know that $$\forall \epsilon>0\quad \exists M\quad \forall n>M\quad |x_n|<\epsilon$$also $|y_n|<B$ for some $B>0$ therefore$$|x_ny_n|=|x_n|\cdot |y_n|=B|x_n|\le B\epsilon$$for $n>M$. Therefore$$\lim_{n\to\infty} x_ny_n=0$$

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