Nearly universal consistency of maximum likelihood in discrete models

The consistency of the maximum likelihood estimator (MLE) has been well studied in many papers such as Wald (1949), Kiefer and Wolfowitz (1956) and many more subsequent works. The purpose of this short note is to provide a new direction to understand the consistency of the MLE in discrete models. In addition, our work gives a very general and direct proof for the consistency of the MLE by introducing a parameter-free version of consistency.