Transpositions of adjacent elements generate the symmetric group on a finite set

Statement
Let $$S$$ be a totally ordered finite set. Then, the set of fact about::transpositions between adjacent elements in the total ordering of $$S$$ generates the fact about::symmetric group on $$S$$.

Note that any totally ordered set can be identified with the set $$\{1,2,3,\dots,n\}$$ with the usual integer ordering, so the above can be rephrased as: the set of transpositions of the form $$(i,i+1)$$ generate the symmetric group on $$\{1,2,3,\dots,n\}$$.

It turns out that this generating set is both Sims-reduced and Jerrum-reduced.

Related facts

 * All transpositions involving one element generate the finitary symmetric group