Prove that $\lim_{n\rightarrow \infty} (x_1+x_2+...+x_n)/n= \infty$ I need help with this exercise:
Consider the sequence $\{x_n\}$ that verifies $$\lim_{n\rightarrow \infty} x_n = +\infty$$
Now prove that $$\lim_{n\rightarrow\infty} \frac{x_1+x_2+...+x_n}n= +\infty$$
Also I need to find an example that the converse does not apply.
Thank you for all your answers.

Let $M>0$. Since $\lim_{n\to\infty}x_n=\infty$, there exists $N>0$ such that
$$ x_n\ge 4M \text{ for }n> N. $$
For fixed $N$, since 
$$ \lim_{n\to\infty}\frac{x_1+x_2+\cdots+ x_{N}}{n}=0 $$
there is $N_1>N$ such that
$$ \frac{x_1+x_2+\cdots x_{N}}{n}+M>0. $$
So, for $n>\max\{2N,N_1\}$, one has
\begin{eqnarray} 
&&\frac{x_1+x_2+\cdots+ x_n}{n}\\
&=&\frac{x_1+x_2+\cdots+ x_{N}}{n}+\frac{x_{N+1}+x_{N+2}+\cdots x_{n}}{n} \\
&\ge&\frac{x_1+x_2+\cdots+ x_{N}}{n}+\frac{n-N}{n}\cdot4M\\
&\ge&\frac{x_1+x_2+\cdots+ x_{N}}{n}+\frac12\cdot4M\\
&\ge&M.
\end{eqnarray}
Thus
$$ \lim_{n\to\infty}\frac{x_1+x_2+\cdots+ x_n}{n}=\infty. $$
