(f) Let \( V \) be a vector space and \( d \) be a metric on \( V \) such that
(i) \( d(x, y)=d(x+z, y+z) \) for all \( x, y, z \in V \), and
(ii) \( d(\lambda x, \lambda y)=|\lambda| d(x, y) \) for all \( x, y \in V \) and \( \lambda \in \mathbb{F} \).
Define \( \|\cdot\| \) by
\[
\|x\|=d(x, 0) .
\]
Show that \( \|\cdot\| \) is a norm on \( V \). Next, show that the metric induced by \( \|\cdot\| \) is precisely \( d \).
