a) Let V be a normed vector space. Then show that (by the triangle inequality) the function f(x)=||x|| is a Lipschitz function from V into [0,∞). In particular, f is uniformly continuous on V.
b) Show that a closed subset F of contains an element of minimal norm, that is, there is an x E F such that ||x||≤||y|| for all y E F. (here ||x|| refers to the usual Euclidean norm).
(hint: F may not be compact, so work on a suitable compact subset of F.)
I proved part a, but I really have no idea how to do part b.
I hope someone can help me out! Thank you!
[also under discussion in math links forum]