Here is the question:
My attempt: We know that any permutation can be expressed as a product of transpositions. Any kind of transposition $(a_{k}, a_{l})$ can be expressed as $(a_{k}, a_{k+1})(a_{k+1}, a_{k+2})...(a_{l-1}, a_{l})(a_{l-2}, a_{l-1})...(a_{k}, a_{k+1})$. So, I try to take the k-th element to the l-th place and then the l-th element to k-th one. The elements between these two elements stay fixed. Is this correct?
This is obvious: you know that the set of all transpositions generates $S_n$. It remains to be seen that any transposition can be expressed as a product of transpositions of the type $(i,i+1)$, and this is exactly the that you wrote above.