## Disproving a Statement By Counter Example

Proofs are often hard. It is often easier to prove that a statement is not true by finding a counterexample.
Example: Is it rue that if
$x \gt y$
then
$x^2 \gt y^2$
?
No it is not. Take
$x=1, \: y=-2$
then
$1 \gt -2 \rightarrow x \gt y$
but
$1^2=1 \lt (-2)^2=4 \rightarrow x^2 \lt y^2$
.
Is it true that if
$x \gt y$
then
$\frac{1}{x} \lt \frac{1}{y}$
?
No it is not. Take
$x=2, \: y=-1$
. Then
$x \gt y$
but
$\frac{1}{2}= \frac{1}{x} \gt \frac{1}{y}=- \frac{1}{2}$