What is the justification of using the first-order Taylor expansion in the proof of asymptotic normality of MLEs?

Question

Proofs of asymptotic normality that I have seen involve taking the first-order taylor expansion of the score function (derivative of the log-likelihood) about the MLE estimate evaluated at the true value of the parameter, and then using the central limit theorem, law of large numbers, and Slutsky's theorem. If one assumes that the MLE is asymptotically unbiased and the score function behaves sufficiently well, then this expansion seems justified.

But asymptotic unbiasedness is one of the results that is proven using asymptotic normality of MLEs, correct? So how do we know a-priori that the MLE is asymptotically unbiased, without using circular logic?

Answer 1

The proofs I am familiar with

a) first prove consistency

b) then use a mean-value expansion, not a Taylor expansion, so they do not have a remainder, and the unknown value at which the Hessian must be evaluated lies between the estimator and the true value - but because we have already proved consistency it will be sandwiched to the true value asymptotically.