Calculate the limit $\frac{(2n-1)x_1+(2n-3)x_2+\dots +3x_{n-1}+x_n}{n^2}$ when $x_n\to x$. 
Let $\{x_n\}$ be a sequence in $\Bbb R$ and $x_n\to x$ as $n\to \infty$.
Then $$\frac{(2n-1)x_1+(2n-3)x_2+\dots +3x_{n-1}+x_n}{n^2}\to x$$.

Do anyone know how to solve this kind of problem efficiently? I think I need to estimate $$\left|\frac{(2n-1)x_1+(2n-3)x_2+\dots +3x_{n-1}+x_n}{n^2}-x\right|$$ Is there some other ways to calculate the limit directly?
Thanks for any comments.

(Fill in the gaps as needed. If you're stuck, explain what you've done and why you're stuck.)

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*Show that if $ x_i \rightarrow x$, then$ \frac{1}{n} \sum x_i \rightarrow x$.


*Show that if $ x_i \rightarrow x$, then $ \frac{1}{n^2} \sum (2i-1) x_i \rightarrow x$.


*Hence, conclude that $ \frac{2}{n} \sum x_i - \frac{1}{n^2} \sum (2i-1) x_i \rightarrow x $
