Hello I need help in this matrix problem. I know it seems pretty simple but I am having hard time proving it..

Let U E M(nxn) (C) be a unitary matrix, that is UU*=U*U=I. Let A, X be eigenpair for U. Prove that |A| = 1

---This is what I have got. So please tell me if I need to add something or is it completely wrong approach. Thanks for the kind help!

Let U be the unitary matrix...

then, Suppose x is an eigenvector corresponding to the eigenvalue λ of U. Then Ux = λx, so

||Ux|| = ||λx|| = |λ| . ||x||

But U preserves lengths, so ||Ux|| = ||x||, and hence |λ| = 1....Do I have to give the reason why U preserves lengths?? and how??