I am having a hard time understanding the way my textbook explains how to check consistency. I understand the concept of how it refers to the shape of the pdf of an estimator as a function of $\displaystyle n$ and the two examples are clear, but as soon as I try an exercise I am lost.

The first one I am looking at is

Let $\displaystyle Y_1, Y_2,...,Y_n$ by a random sample of size $\displaystyle n$ from a normal pdf having $\displaystyle \mu=0$. Show that $\displaystyle S_n^2=\frac{1}{n}\sum_{i=1}^nY_i^2$ is a consistent estimator for $\displaystyle \sigma^2=Var(Y)$

I tried this:

$\displaystyle \lim_{n\to\infty}P(\frac{1}{n}\sum_{i=1}^nY_i^2-\sigma^2<\epsilon)=1$

I am not sure how to write this probability though. I tried to write it as an integral, but then I don't know how to solve the integral I got. I am really just guessing at this point, so if I could have any advice that would be great.