Permutation/Sign via transpositions/Fact

Let ${}M=\{1,\ldots ,n\}$ and let ${}\pi$ be a permutation on ${}M$. Let

${}\pi =\tau _{1}\cdots \tau _{r}\,$

be written as a product of ${}r$ transpositions.

Then the

sign can be described as

${}\operatorname {sgn} (\pi )=(-1)^{r}\,.$

Proof, Write another proof