Prove key of leaf is larger or smaller than key of parent if leaf is largest key in smaller tree or smallest in larger tree with respect to parent
In Order traversal sorts key in an ascending order. If $u$ is the parent of leaf $v$ then $v$ is either left or right child of $u$. If $v$ is left child then $key[v]\leq key[u]$ which implies $key[v]$ is the largest key in the tree smaller than $key[u]$. Similar intuition follows if $v$ is right child.