Show that if $\displaystyle lim(\frac{a_{n}}{n}) = L$ for $\displaystyle L>0$, then $\displaystyle lim(a_{n}) = + \infty$

Here's what I did:

$\displaystyle lim(\frac{a_{n}}{n}) = \frac{lim(a_{n})}{lim(n)} = L $

Then $\displaystyle \\ lim(a_{n}) = L \cdot lim(n) $

And since $\displaystyle \\ lim(n)$ goes to $\displaystyle +\infty$ and $\displaystyle L$ is a constant greater than 0,

the $\displaystyle lim(a_{n})$ also goes to $\displaystyle + \infty$

However, I'm pretty sure I need to use the definition of convergence to proves this, but I'm not sure how. Thank you.