Show that if for , then
Here's what I did:
Then
And since goes to and is a constant greater than 0,
the also goes to
However, I'm pretty sure I need to use the definition of convergence to proves this, but I'm not sure how. Thank you.
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 .