I think it is right but you did it so messed up and unclear that, apparently, you succeeded to confuse, and perhaps even annoy, your instructor.
After proving the claim for n=1, you must say/write: let us assume the claim's true for n (this is also called "inductive hypothesis") and we shall prove for n+1, i.e. we must prove that
. We now take the LHS and develop it:
and we're done since now we got the RHS...