Every Permutation is a Product of Adjacent Transpositions

Let $n\in\mathbb{N}$, and let $\sigma\in S_n$ be a permutation in the sense of Permutation of the Set $\{1,\dots,r\}$. For each $r\in\{1,\dots,n-1\}$, define the adjacent transposition $\tau_r\in S_n$ by

and $\tau_r(m)=m$ for every $m\in\{1,\dots,n\}\setminus\{r,r+1\}$. Then there exist an integer $N\ge 0$ and indices $r_1,\dots,r_N\in\{1,\dots,n-1\}$ such that