The Proof of the Inscribed Square Problem using Topological Degree

We present a proof demonstrating the existence of a non-degenerate inscribed square in any C⁰ Jordan curve in the plane. The main idea is to formulate the square condition based on distances, constructing a continuous map F: T⁴⁴ for any continuous curve. By uniformly approximating the C⁰ curve by a sequence of C¹ curves, we show that the corresponding maps Fₙ converge uniformly to F₀. A detailed analysis of the boundary T⁴ confirms that all Fₙ are non-zero on the boundary, and a uniform positive margin exists. Furthermore, the sequence of zero points does not approach the boundary, thus excluding degenerate squares. Combining these elements, we extend the argument of topological degree from the C¹ case to the C⁰ curve, resolving Toeplitz's conjecture for continuous curves.