How to prove that $\lim_{n \to \infty} \frac{3^n-1}{2 \cdot 3^n} = \frac{1}{2}$? I used this limit as an argument in a proof I wrote (Proof by induction that $\sum\limits_{k=1}^n \frac{1}{3^k}$ converges to $\frac{1}{2}$). 
I was told I should "prove" the limit but given no indication as to how to go about it. I didn't even know it was possible to formally prove a limit, but if it is I'd love to know how to do it.

Hi it's another way to calculate $S=\Sigma_{i=1}^{\infty}\frac{1}{3^i}$
Just multiply 3 at both side, then you will obtain
$3S=\Sigma_{i=1}^{\infty}3*\frac{1}{3^i} = 1+\Sigma_{i=1}^{\infty}\frac{1}{3^i}$
Do substraction, easy to get $S=\frac{1}{2}$
