Hmm. You have
The RHS is
So you are correct in that it does boil down to whether
That inequality is most definitely false. Counterexample:
Then let's suppose We'd have
For positive this does not hold. Therefore, I'd say this inequality of the proof is invalid, and you were right to doubt it.
See here for a valid proof.
Your proof, unfortunately, doesn't work because the expression
doesn't parse mathematically. The result of a dot product is a scalar. The dot product of two scalars is not defined (unless you mean ordinary multiplication in the real numbers).
You should also be suspicious of your proof, because you've proven too much. Is it really true that
Plug in two orthonormal vectors, and you can see that the equality won't work. The LHS is 1, and the RHS is zero.