Al right, I’m reading Baby Rudin on my own, and I’m going to share the last proof I’ve understood. It’s in the second chapter: topology.

As usual this proofs in the book comes with the absolute minimum explanation, so here I’m going to add my explanations.

**Theorem** Every neighborhood is an open set.

*‘So here the first difficulty for me was to grasp that once we
pick a neighborhood
, for any point
inside of this neighborhood we need to show that we can
form another neighborhood around
that is fully contained in the first (bigger)
neighborhood.’*

**Proof** Consider a neighborhood
, and let
be any point of
. Then there exists a positive real
such that:
For all points
such that
*‘this is the second neighborhood around q’*, we
have then
So that
Thus
is an interior pont of E.