Showing ||x|| ≥ ||xiei|| for basis vectors in normed spaces

[Oster]

Statement: V is a finite dimensional vector space with basis {ei} (i goes from 1 to n). V has a norm || || defined on it(not necessarily induced by an inner product). Let x=Ʃxiei belong to V. I want to show that ||x|| ≥ ||xiei|| for any fixed i.

I'm not entirely sure this result is correct. But i remember seeing something similar in a text a while ago.
I know all the properties of a norm but I'm not sure how to proceed. I don't know how the independence of the basis vectors will fit into the proof.

 

[LCKurtz]

It isn't true. In $R^2$ consider the basis $e_1=<1,0>,\ e_2=<-1,.1>$. Then let $x=1e_1+1e_2=<0,.1>$. Then $\|x\|=.1<1\|e_1\|$.

 

[Oster]

Thank you!