This paper aims to address a brief concept of Metric Spaces in order to present and
demonstrate Banach’s Fixed-Point Theorem in Euclidean Space-ndimensional R
n
. In
addition, display some of its applications, especially in the demonstration of Existence and
Uniqueness of Solutions of Ordinary Differential Equations. For this, we had to first study
essential concepts such as the Topological Aspects of Space R
n
, Metric Spaces and Spaces
Complete Metrics, in order to define Banach Spaces and then to enunciate, demonstrate
and show some applications of the Banach’s Fixed-Point Theorem for contractions which
tells us that if R
n
is a complete metric space, every contraction f : R
n → R
n has a single
fixed point in R
n
, that is, there exists a unique x ∈ R
n
such that f(x) = x.
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