A Discussion of Arnold's Limit Problem and its Geometric Argument

TL;DR

This paper re-evaluates Arnold's limit problem, confirming the lemma's correctness with power series methods and identifying a flaw in the original geometric proof through a counterexample.

Contribution

It provides a new proof of Arnold's lemma using power series and highlights a defect in the original geometric argument.

Findings

Confirmed the correctness of Arnold's lemma using power series

Constructed a counterexample showing flaw in the original geometric proof

Clarified the geometric argument's limitations

Abstract

Upon re-examining Arnold's established lemma for explaining his famous limit problem, we have determined that while the lemma itself is correct, there is a defect in the original geometric proof. In this paper, we prove the correctness of the lemma using methods of power series, and construct a counterexample to illustrate the defect in Arnold's geometric proof.