Every AVL tree may be red black tree I proved by induction that every AVL tree may be colored such that it will be red black tree. The problem is that I can't see an error in my proof.  Look at my proof.

Induction for height.
Let's assume that it is truth for each AVL tree of height at most $h$.
Let's consider AVL tree $T$ of height $h+1$.  Now, let's consider two subtree of $T$ - $L$ and $R$. We know that $height(R)\le h$ and $height(L)\le h$.   Hence using induction hypothesis we conclude that $L$ and $R$ may be colored such that $L$ and $R$ will be red black tree.  Then we may paint root - of course black color.   Now $T$ is AVL and black tree.
Your proof produces a tree in which all nodes are colored black. It doesn't necessarily satisfy the "black height" rule:


  Every path from a given node to any of its descendant NIL nodes contains the same number of black nodes.


Not every AVL tree satisfies this condition, for example the Wikipedia example doesn't.