Real analysis: norms

kingwinner

"Let V be a normed vector space. Then 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."

Show that a closed subset F of R^n 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.)

請高手賜教, thx!

Naozumi

呢題我只係解決左一部份,F係finite,同F有limit point時的case, F係infinite但冇limit point個case未諗到點解決