How to prove that $d(v,w)=0 \iff v=w$ [closed]
Let $V$ a vector space and $(p_{j})^{\infty}_{j=1}$ a collection of seminorms defined on $V$ with the property that if $p_{j}(v)=0$ for all $j$, then $v=0$. Prove that $d(v,w)=\sum_{j=1}^{\infty}\frac{1}{2^j}\frac{p_{j}(v-w)}{1+p_{j}(v-w)}$ defines a distance in $V$
Hint: If $a_j \ge 0$ for all $j$, then $\sum_j a_j =0 \iff a_j =0$ for all $j$.