Help Needed in Proof

Swapity Swap

Let suppose, initial position of element was A , and the position where it starts to repeat after some number of operation is B.
Now if we solve problem by hand and observe the pattern,the situtation is always like this
but why it can’t be like this
So , is there any way we can proof that 2nd secenerio will never happen ??

A permutation can always be decomposed into cycles.

Thnx … Got it.