These are the first steps in a simple sketch of the proof that, under reasonable conditions, the max likelihood estimate is consistent.
I follow the steps all the way down to entering the true parameter value for the estimate. How can you enter the "true parameter" constant into the equation before taking the derivative and then move derivative outside the integral? Obviously, the integral was going to give a constant and then the derivative was going to be 0. Would this not have happened for any constant entered for the parameter variable?
In the same way if I want to show 0 is a critical value for x^2, I cannot enter 0^2=0, then take derivative of 0, show that this is 0 and say I found a critical value. Taking derivative of 2^2=4 would have also been 0 but 2 is not a critical value in this case.
