The Cesàro Mean Theorem in the infinite case. I am trying to prove the Cesàro Mean Theorem in the infinite case. Let me state my problem more precisely.

Problem. Let $ (a_{n})_{n \in \mathbb{N}} $ be a sequence in $ \mathbb{R} $ such that $ \displaystyle \lim_{n \to \infty} a_{n} = \infty $. Then prove that
  $$
\lim_{n \to \infty} \frac{1}{n} \sum_{k = 1}^{n} a_{k} = \infty.
$$

I tried to use only the definition of convergence, but I got stuck. I know that the result is an immediate consequence of the Cesàro-Stolz Theorem, but I am looking for a solution using the definition of convergence alone. Any suggestions? Thanks to all helpers!

Since $a_{n}\to\infty$ as $n\to\infty$ then there exists $N$ such that if $n\ge N$ then $a_{n}\ge M>0$. This means at most finitely many terms are negative. By choosing $N$ large enough we may also assume that $\sum_{k=1}^{n}a_{k}>0$ for $n\ge N$. By perhaps choosing $n$ even larger we may assume that $\lvert\frac{\sum_{k=1}^{N}a_{k}}{n}\rvert<1$ So:
$\frac{\sum_{k=1}^{n}a_{k}}{n}=\lvert\frac{\sum_{k=1}^{n}a_{k}}{n}\rvert\ge\frac{\sum_{k=N+1}^{n}a_{k}}{n}-\frac{\sum_{k=1}^{N}a_{k}}{n}>\frac{n-N-1}{n}M-1=(M-1)-\frac{N+1}{n}$
