Simple Inequality Proof Seems Baffling So, a Real Analysis question I've found states:

For a,b,c $\in\mathbb{R}$, prove: ( $a\leqslant b$ and $c\leqslant 0$) $\implies$ ($ac\leqslant bc$)

But immediately of course this seems opposite to what it should imply, and you can think of any number of counterexamples to disprove it: 
Let $a = 3$, $b = 7$, and $c = -1$.
Then, $a\leqslant b$ and $c\leqslant 0$, but $ac = -3$ and $bc =-7$, thus $ac\geqslant bc$.
Have I lost the plot, or has the question been written in error?
Thanks, much appreciated.

The stated property is wrong. It should be
$$(\text{ $a\leqslant b$ and $c\leqslant 0$) $\implies$ ($ac\geqslant bc$)}.$$
Note that $a\leq b$ and $c\leq 0$ imply that $b-a\geq 0$ and $-c\geq 0$. Now recall that the product of two non-negative numbers is non-negative. Hence $(b-a)(-c)\geq 0$, that is $ac\geq bc$.
