Demonstração de dois teoremas sobre Sequências de Intervalos Encaixantes

Abstract

In a widely used work in the teaching of Calculus in Higher Education Institutions in Brazil, a reference is made to the
Property of Nested Intervals to demonstrate the existence of real roots for a quadratic equation, as well as to prove the
existence of roots for equations involving any powers of real unknowns. It is mentioned that if a sequence of intervals is
nested, then a sequence formed with the square or with any integer power of its terms is also nested, although a proof of
these claims is not presented. In this paper, it is demonstrated that if the Property of Nested Intervals holds for a sequence
of intervals, then it also holds for the sequence formed by the square or any integer power of its terms. The proof is based
on the validity of two Properties that underlie the Property of Nested Intervals and has been divided into two cases to include
all possibilities. The conclusion is based on the deduction inferred from the proofs of the fundamental properties, filling
this gap for a better understanding of the subject.

Keywords: Existence of real roots. nested intervals sequences. nested intervals.