Ordering of Real Numbers is Reversed by Negation

Theorem

Let $x$ and $y$ in $\R$ be real numbers such that:

$x \le y$

Then:

$-y \le -x$

where $-y$ and $-x$ are the negatives of $y$ and $x$ respectively.

Proof

By definition of ordering:

$x \le y$

if and only if:

$x < y \text { or } x = y$

From Order of Real Numbers is Dual of Order of their Negatives:

$x < y \iff \paren {-x} > \paren {-y}$

Hence the result.

$\blacksquare$