What you have might serve as some intuition, but it does not work as a proof. You used the limit laws to rearrange the expressions, but they only work when the corresponding limits exist. Here is a slightly more formal argument:
Since , we know that given any , when is sufficiently large, we have . Multiplying through by gives . Breaking the absolute values and rearranging, .
This shows that as long as we pick and take , we have , so .