Ergodic theory is the study of the long term average behavior of systems evolving in time. Due to its inherent complexity, it may be infeasible to obtain analytical solutions to dynamical systems. One way to study these dynamical systems is to consider its ergodic properties. After giving a brief introduction to ergodic theory, I will discuss how the weak-ergodicity of discrete time, time-inhomogeneous Markov chains is related to maximizing a seminorm, based on the Markov kernel, over a convex polytope. Knowledge of probability helpful.