LaTeX note: use \mathcal, not \cal, in formulas (\cal works globally, and converts the rest of the formula to gibberish).

Use the theorem which says that the boundary of the spectrum is contained in the approximate point spectrum. Suppose that is an approximate eigenvalue for T, with approximate eigenvector x. Then x is a unit vector, and . But , and it follows that .

Thus . It then follows on purely geometrical grounds that either or . The first alternative occurs when (in other words, when T is unitary), and the second alternative in the case when (in other words, when T is a proper isometry).