Eigenvalues of real symmetric matrices

TL;DR

This paper provides a novel proof demonstrating that real symmetric matrices always have real eigenvalues, using only basic concepts like supremum and infimum, avoiding traditional limit or compactness methods.

Contribution

It introduces a new proof technique for eigenvalue existence that relies solely on supremum and infimum, simplifying the theoretical framework.

Findings

Real symmetric matrices have real eigenvalues.

A proof using only supremum and infimum is established.

Traditional limit or compactness arguments are avoided.

Abstract

We present a proof of the existence of real eigenvalues of real symmetric matrices which does not rely on any limit or compactness arguments, but only uses the notions of "sup", "inf".