Fix the child at position $i$, initially holding $k$ chocolates. Let $s$ be the number of children to their left who initially held fewer chocolates.

There are $k-1$ smaller amounts in total, so this child gives away $k-1-s$ chocolates and keeps

$$k-(k-1-s)=1+s.$$

They give at most $k-1$ chocolates, so all gifts can be made from their initial supply.