# $\left\lbrace{s_{k}}\right\rbrace_{k\in\mathbf{N}}$ is convergent on $R^n$, with ${s_{k}}\rightarrow \langle a,b\rangle$

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

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

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