Prove that if $C$ is a convex set containing $B(r)$, then $\sup\{d(y,0)\mid y\in C\}=\infty$ Let $0<p<1$. Define a metric on $l^p$ by $d((a_k)_{k=1}^\infty,(b_k)_{k=1}^{\infty})=\sum_{k=1}^\infty |a_k-b_k|^p$. For any $r>0$, let $B(r)=\{x\in l^p\mid d(x,0)<r\}$. Prove that if $C$ is a convex set containing $B(r)$, then
$\sup\{d(y,0)\mid y\in C\}=\infty$. Deduce that $l^p$ is not a locally convex topological vector space. 
How to prove this question? Thanks.

Hint: denoting by $e_n$ the $n$-th vector of "canonical basis" of $\ell^p$, compute $d(x_N, 0)$ for each $N\in\mathbb N$, where $x_N=\frac 1N \sum_{n=1}^N e_n$.
