  The Glivenko-Cantelli theorem states that the empirical distribution function
converges uniformly almost surely to the theoretical distribution for a random
variable $X \in \mathbb{R}$. This is an important result because it establishes
the fact that sampling does capture the dispersion measure the distribution
function $F$ imposes. In essence, sampling permits one to learn and infer the
behavior of $F$ by only looking at observations from $X$. The probabilities
that are inferred from samples $\mathbf{X}$ will become more precise as the
sample size increases and more data becomes available. Therefore, it is valid
to study distributions via samples. The proof present here is constructive,
meaning that the result is derived directly from the fact that the empirical
distribution function converges pointwise almost surely to the theoretical
distribution. The work includes a proof of this preliminary statement and
attempts to motivate the intuition one gets from sampling techniques when
studying the regions in which a model concentrates probability. The sets where
dispersion is described with precision by the empirical distribution function
will eventually cover the entire sample space.
