Find the limit $\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}$ When $\lim_{n\to\infty} x_n = a$, and $\lim_{n\to\infty} y_n = b$, find the limit,
$$\lim_{n\to\infty} \frac{x_1 y_n + x_2 y_{n-1} + \cdots + x_n y_1}{n}.$$
Thank you for your help in advance.

We can use the standard result:

If $x_{n} \to x$ as $n \to \infty$ then $$\lim_{n \to \infty}\frac{x_{1} + x_{2} + \cdots + x_{n}}{n} = x$$

This is pretty standard and its proof is available on MSE. Now for the current question let $x_{n} = a + e_{n}$ and then $e_{n} \to 0$ as $n \to \infty$. We have $$\begin{aligned}\frac{x_{1}y_{n} + x_{2}y_{n - 1} + \cdots + x_{n}y_{1}}{n} &= a\cdot\frac{y_{1} + y_{2} + \cdots + y_{n}}{n}\\
&+ \frac{e_{1}y_{n} + e_{2}y_{n - 1} + \cdots + e_{n}y_{1}}{n}\end{aligned}$$ Now in the above the first term tends to $a\cdot b = ab$. For the second term we need to note that the sequence $y_{n}$ is bounded by some $K > 0$ and therefore in absolute value the second term is no greater than $K\cdot\dfrac{|e_{1}| + |e_{2}| + \cdots + |e_{n}|}{n}$. Since $e_{n} \to 0$ therefore this term also tends to $0$. So the desired limit is $ab$.
