We want to order the real numbers.1
Suppose $R, S \subset \R $. If $R \subset S$ and $R \neq S$ then we say that $R$ is less than $S$ If $R \subset S$ then we say that $R$ is less than or equal to $S$. This can be shown to define a total order on $\R $.
If $R$ is less than $S$ we write $R < S$. If $R$ is less than or equal to $S$ we write $R \leq S$.