# The Transitivity of Whole Numbers

Introduction

The less than relation is an order relation of real numbers.  In this post, however, to prove one of its properties, we limit it to the set of whole numbers.  In the set of whole numbers, $a$ is less than $b$, or $a < b$ in symbol, if and only if there exists a whole number $m$ such that $a + m = b$. For example, we are sure that $9 < 11$ because there exists a whole number $2$ such that $9 + 2 = 11$. Below we prove one of the properties of order relations in the set of whole numbers.  That is, if $a < b$ and $b < c$, then $b < c$. This is called the transitivity property for the relation $<$.

Theorem

If $a < b$ and $b < c$, then $a < c$.

Proof

From the definition above, $a < b$ if and only if there exists a whole number $p$ such that $a + p = b$.

It follows that $b < c$  means there exists a whole number $q$ such that $b + q = c$.

By substitution, $(a + p) + (b + q) = b + c$.

Subtracting $b$ from both sides of the equation and regrouping gives us $a + p + q = c$.

If, we let $r = p + q$, then $r$ is also a whole number since $p$ and $q$ are whole numbers.

By substitution, we have $a + r = c$.

Since there exists a whole number $r$ such that $a + r = c$, it follows that $a < c$ $\blacksquare$.