Gifts to the Right 500 pts

There are $n\ge 1$ children in a row, initially holding $1,2,\ldots,n$ chocolates in some order.

Each child gives one chocolate to every child to their right who initially held fewer chocolates. All comparisons use the initial amounts.

Prove that after these gifts, the $i$-th child from the left has exactly $i$ chocolates for every $1\le i\le n$.