Definition:Less Than (Real Numbers)

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Let $\R_{>0}$ denote the set of strictly positive real numbers.

Let $x, y \in \R$.

Then we write $x < y$ if and only if:

$ y - x \in \R_{>0}$

and we say that $x$ is less than $y$.

Also see

Inequality iff Difference is Positive, where it is shown that this may be deduced from the Ordering Properties of Real Numbers.