Short proofs for cut-and-paste sorting of permutations

We consider the problem of determining the maximum number of moves required to sort a permutation of [n][n] using cut-and-paste operations, in which a segment is cut out and then pasted into the remaining string, possibly reversed. We give short proofs that every permutation of [n][n] can be transformed to the identity in at most ⌊2n/3⌋⌊2n/3⌋ such moves and that some permutations require at least ⌊n/2⌋⌊n/2⌋ moves.