This post is part of a series which answers a certain question about the supremum of a Gaussian process. I am going to write, as I have understood, a proof given in Chapter 1 of the book “Generic Chaining” by Michel Talagrand. I recommend the reader to take a look at the excellent posts by James Lee on this matter. (I am a beginner, James Lee is a master.)

Let be a finite metric space. Let be a Gaussian process where each is a zero-mean Gaussian random variable. The distance between two points is the square-root of the covariance between and . In this post, we are interested in upper-bounding .

**Question:** How large can the quantity be?

In this post we are going to prove the following fact:

where is the distance between the point from the set , and is a specific sequence of sets with . Constructions of these sets will be discussed in a subsequent post.