On the history of nested intervals: from Archimedes to Cantor

The idea of the principle of nested intervals or the concept of convergent sequences, which is equivalent to this idea dates back to the ancient world. Archimedes calculated the unknown in excess and deficiency, approximating with two sets of values: ambient and nested values. J. Buridan came up with a concept of a point lying within a sequence of nested intervals. P. Fermat, D. Gregory, I. Newton, C. MacLaurin, C. Gauss, and J.-B. Fourier used to search for an unknown value with the help of approximation in excess and deficiency. In the 19th century, in works of B. Bolzano, A.-L. Cauchy, J.P.G. Lejeune Dirichlet, K. Weierstrass, and G. Cantor, this logical construction turned into the analysis argumentation method. The concept of a real number elaborated in the 1870s in works of Ch. Méray, Weierstrass, H.E. Heine, Cantor, and R. Dedekind. Cantor’s elaboration based on the notion of a limiting point and principle of nested intervals. What we are going to consider now, is the genesis of this idea, which dates back to the ancient world.