Let's suppose without loss of generality
that ||x|| is no smaller than
||y||. (Otherwise we just interchange the roles
of x and y.)
Thus we have to show that
This follows directly from the triangle inequality itself if we write x as
and think of it as
x=(x-y) + y.
Taking norms and
applying the triangle inequality gives
which implies (*).
Fine print, your comments, more links, Peter Alfeld, PA1UM.