O uso do conceito de séries para demonstrar a igualdade “0,999...=1”

Abstract

This work aims to rigorously present a justification for the equality 0.999 . . . = 1, which often causes confusion among students and mathematics enthusiasts. It is worth noting that this equality is related to the very history of the construction of the real numbers and the concept of infinity in mathematics. It was through the work of Augustin-Louis Cauchy (1789−1857) and Karl Weierstrass (1815−1897) that the foundations of calculus and real analysis were consolidated, allowing the equality 0.999 . . . = 1 to be treated precisely. In the 20th century, with the formalization of the real numbers through Cauchy sequences or Dedekind cuts, it became clear that this equality refers to a sequence that converges (or “approaches”) 1.