if $\{x_n\}$ and $\{y_n\}$ converge, then $$\lim_{n \to \infty} \frac{1}{n} \bigg(\sum_{k=1}^n x_ky_k\bigg)=\lim_{n \to \infty}x_n \lim_{n \to \infty}y_n$$
Let's say that $\{x_n\} \to x$ and $\{y_n\} \to y$.
Intuitively, I can see that for a very large $n$, the summation will look very similar to $\frac{nxy}{n}=xy$. However, I can't seem to find a proper proof for this (how could I translate this intuitive thought into a formal proof?). I tried using the fact that the sequences are bounded but that leaves me with
$$ \inf (x_n)\inf(y_n)\le\lim_{n \to \infty} \frac{1}{n} \bigg(\sum_{k=1}^n x_ky_k\bigg)\le\ \sup (x_n)\sup(y_n)$$
I'd love to see some hints. By the way, I'm only using basic aspects about sequences: Cauchy, Weierstrass theorem, Definition of limits, Squeeze theorem and so on. That is, no derivatives or integrals.
Given $\epsilon>0$ choose $N\in\Bbb N$ such that $k>N\implies|x_ky_k-xy|<\epsilon$. $$\big|\lim_{n\to\infty}\big(\frac{1}{n}\sum_{k=1}^nx_ky_k\big)-\lim_{n\to\infty}x_ny_n\big|=\big|\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^n(x_ky_k-xy)\big|\leq\lim_{n\to\infty}\frac{1}{n+N}\sum_{k=1}^{n+N}|x_ky_k-xy|$$ $$\leq\lim_{n\to\infty}\frac{1}{n+N}(n+N)\epsilon=\epsilon$$ $$\therefore\;\lim_{n\to\infty}\frac{1}{n}\sum_{k=1}^{n}|x_ky_k-xy|=xy=\lim_{n\to\infty }x_ny_n$$