Convergence implies convergence of norms
If x_n→x, then ||x_n||→||x||
Proposition. If converges in norm to in a normed space, then
Context. This expresses continuity of the norm map .
Proof sketch. By the reverse triangle inequality,
hence .
Let be a normed vector space.
A sequence in converges to (in norm) if
Equivalently: for every there exists such that for all .
Context. By the metric induced by a norm, this is exactly convergence in the associated metric space.