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