Preliminary investigations on induction over real numbers

TL;DR

This paper introduces a novel induction principle applicable to real numbers, extending the classical natural number induction to a continuous domain, which could impact analysis and related fields.

Contribution

It proposes a new induction principle for real numbers, filling a gap in the theoretical framework for reasoning over continuous domains.

Findings

Formulated a real number induction principle

Established theoretical foundations for real induction

Potential applications in analysis and formal verification

Abstract

The induction principle for natural numbers expresses that when a property holds for some natural number a and is hereditary, then it holds for all numbers greater than or equal to a. We present a similar principle for real numbers.