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 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 by a random sample of size from a normal pdf having . Show that is a consistent estimator for

I tried this:

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.