More Proof by induction(the rearranging bit)

Keep in mind that would give you the $U_{k+1}$. For $U_k$ you'd need to map $k \mapsto (k-1)$ on the right side. This would give you the exact same thing as mine would and proof would hold.

The proof for showing the results for $U_k$ or $U_{k+1}$ are equivalent, they're exactly the same. With induction all you need to show is that if the result is true for $U_k$ then using logical arguments it implies that the result is also true for $U_{k+1}$.