Consider the definition of a norm on a real vector space X. I want to show that replacing the condition
$\|x\| = 0 \Leftrightarrow x = 0\quad$ with $\quad\|x\| = 0 \Rightarrow x = 0$
does not alter the the concept of a norm (a norm under the "new axioms" will fulfill the "old axioms" as well).
Any hints on how to get started?