An intuitive proof of the Dvoretzky-Hanani theorem in R^2

TL;DR

This paper provides an intuitive proof of the Dvoretzky-Hanani theorem in R^2, demonstrating that divergent series with terms tending to zero can be manipulated to converge through a specific construction.

Contribution

It introduces a novel, intuitive construction-based proof for the Dvoretzky-Hanani theorem in R^2, with extensions proposed for higher dimensions.

Findings

The proof shows how to choose signs to make the series converge in R^2.

The construction provides a new perspective on divergence and convergence in finite-dimensional spaces.

Extensions to R^n are discussed for broader applicability.

Abstract

The Dvoretzky-Hanani theorem states that the general term of any perfectly divergent series in a finite dimensional space does not tend to zero. An intuitive proof is provided R2 using a construction that allows us to determine a choice of +/- such that $$a_1 +/- a_2 +/- a_3 +/- a_4... +/- a_n...$$ converges to a point in the space if ||a_i|| goes to 0. Extensions to the construction are proposed for the general R^n.