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Let $\left\lbrace{a_{k}}\right\rbrace_{k\in\mathbf{N}}$ and $\left\lbrace{b_{k}}\right\rbrace_{k\in\mathbf{N}}$ two convergent sequences in ${R}^n$, with ${a_{k}}\rightarrow a$ and ${b_{k}}\rightarrow b$. Let $s_{k}:=\langle a_{k},b_{k}\rangle$ $\forall_{k\in\mathbf{N}}$. Show that the sequence $\left\lbrace{s_{k}}\right\rbrace_{k\in\mathbf{N}}$ is convergent on $R$, with ${s_{k}}\rightarrow \langle a,b\rangle$.

I think they are using scalar product, but I don't know how to apply that definition to sequences, any idea is very helpful, thanks

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    $\begingroup$ Is $\langle -,-\rangle$ an inner product? In that case, $s_k$ should be a sequence in $\mathbb R$. $\endgroup$
    – Kenta S
    May 19, 2021 at 15:27
  • $\begingroup$ Yes, thank you very much it was a mistake, I already corrected it $\endgroup$ May 19, 2021 at 15:36

1 Answer 1

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For all $k \in\mathbb N$, you have : \begin{align} |\langle a_k,b_k \rangle - \langle a,b\rangle|&\leq |\langle a_k, b_k - b\rangle| + |\langle a_k-a,b\rangle| \end{align}

Since $(a_k)$ is convergent, it is bounded. Let $M>0$ be such that $\forall k\in\mathbb N, \| a_k\| \leq M$

Then, the Cauchy-Schwartz equation gives : $$|\langle a_k,b_k \rangle - \langle a,b\rangle| \leq M \|b_k - b\| + \| a_k - a\| \|b\|$$

Therefore, $$\lim_{k\to \infty} \langle a_k,b_k\rangle = \langle a,b\rangle$$

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  • $\begingroup$ Thank you very much $\endgroup$ May 19, 2021 at 15:43
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    $\begingroup$ @TatianaMalespínAlvarado If this answered the question for you, you should mark the "answered" check box to accept it. This both marks the question as answered in the system as well as awarding some reputation for the person who wrote it. $\endgroup$
    – Alan
    May 19, 2021 at 16:04

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