Proving the existence of the $n$th root by induction

TL;DR

This paper provides a formal proof, using mathematical induction, that every positive real number possesses an $n$th root for any positive integer $n$, establishing a fundamental property of real numbers.

Contribution

It offers a rigorous inductive proof of the existence of $n$th roots for positive real numbers, which is a foundational result in real analysis.

Findings

Confirmed the existence of $n$th roots for all positive real numbers

Provided a formal inductive proof for this property

Strengthened the theoretical basis of root existence in real numbers

Abstract

In this paper we prove by induction on $n$ that any positive real number has $n$th root.