**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, $latex a$ is less than $latex b$, or $latex a < b$ in symbol, *if and only if *there exists a whole number $latex m$ such that $latex a + m = b$. For example, we are sure that $latex 9 < 11$ because there exists a whole number $latex 2$ such that $latex 9 + 2 = 11$.

Below we prove one of the properties of order relations in the set of whole numbers. That is, if $latex a < b$ and $latex b < c$, then $latex b < c$. This is called the transitivity property for the relation $latex <$. Continue reading