Denote $X$, the space of all sequences $\in$ $\mathbb R$. I have a metric $d(x,y):=\sum_{n=1}^\infty 2^{-n}\frac{| x_n-y_n|}{1+| x_n-y_n|}$
and $(X,d)$ is a metric space.
How would I show that the closed (metric) ball $B[0,\frac{1}{3}]$ with centre $0$ and radius $1/3$ is not convex?
I tried working from the definition that a set $A$ in a vector space $X$ is convex if for all $x,y \in A$, and all $t$ in the interval $[0,1]$, $(1 − t ) x + t y \in A,$ but I cant seem to show its not convex.
I was looking up on this, and I read somewhere it might have something to do with summable sequences like $x=(1/2,1/2,1/2,...)?$ Im not sure though, I might be wrong.