real analysis - Arithmetic mean sequence

If the ssequence of $a_n$ goes to infinity as $n$ goes to infinity, then does $$\lim_{n\to\infty} \frac{a_1 + \dots + a_n}{n} = \infty?$$

I know this sequence converges to a finite value if the sequence $a_n$ converges to a finite value, but I don't know if that helps. I've tried using the definition a sequence converging to infinity, I've also tried using the convergence of $\frac{1}{a_n}$ to $0$ to show $\frac{n}{a_1 + \dots + a_n}$ converges to $0$, but no luck. Do I utilize the arithmetic mean inequality? Any hints are more than welcome (only hints please!).

Answer

Indeed,

THEOREM If the limit of $a_n$ goes to infinity as $n$ goes to infinity, then $$\lim_{n\to\infty} \frac{a_1 + \dots + a_n}{n} = \infty$$

PROOF Let $\ C>0.\ $ There exists natural $\ N_C\ $ such that

$\ a_k>2\cdot C\ $ for every $\ k>N_C.\ $ Let

$$ B_C\ :=\ \sum_{n=1}^{N_C}\ a_n $$
and let natural $\ m_C\ $ satisfy

$$ m_C\ >\ N_C-\frac {B_C}C $$

so that
$$ B_C+2\cdot C\cdot m_C\ >\ (N_C+m_C)\cdot C $$

Now, let $\ n>N_C+m_C.\ $ Then

$$ \frac{a_1 + \dots + a_n}n\,\ >
\,\ \frac{B_C\ +\ 2\cdot C\cdot m_C\ +\ \sum_{k=N_C+m_C+1}^n a_k}n $$

$$ >\,\ \frac{(N_C+m_C)\cdot C\,\ +\,\ (n-(N_C+m_C))\cdot2\cdot C}n
\,\ >\,\ C $$

Since $\ C>0\ $ is arbitrary, the theorem holds.   Great!