Modes of Convergence

the sequence after N is always within ε of the supposed limit a. In contrast, the notion of convergence becomes somewhat more subtle when discussing convergence of functions. In this note we briefly describe a few modes of convergence and explain their relationship. Since the subject quickly becomes very technical, we will state many of the fundamental results without proof. Throughout this discussion, fix a probability space Ω and a sequence of random variables (Xn)n∈N. Also, let X be another random variable.