Let X be a normed space Show that the function f X rightarro

Let X be a normed space. Show that the function f: X rightarrow R defined by f(x) = ||x|| is continuous.

Let (X,)(X,) be a normed space. We need to prove that:

(xn):NX xnx impliesxnx

Let >0 and (xn) be an arbitrary sequence in X that converges to xX.

Then,NN:nN implies xnx<

But

|xnx|xnx by the triangle inequality.

Thus,

NN:nN implies Ixnx|<

f(x)=IIxII is continuous