Let S be the set of all sequences $x = \{x_n\}$ of real numbers such that only a finite number of the $x_n$ are $\ne 0$. Define $d(x,y) = max |x_n - y_n|$. Is the space complete? Show that the $\delta$-neighborhoods are not totally bounded.
It’s from Complex Analysis of Lars V. Ahlfors.
I think I can prove it to be complete by showing that every Cauchy sequence converges. But why is it not totally bounded? I have a thought but I don’t know how to write it down. The $d(x,y)$ is a real number so I think the$\delta$-neighborhood should be exactly like a disc in $\mathbb{R}^\infty$, thus we can not measure its volume and do the same estimate like it’s in the finite dimension space. But how can I prove it?
This space is not complete. The sequence$$\left(1,\frac12,\frac13,\ldots,\frac1n,0,0,\ldots\right)_{n\in\mathbb N}$$is a Cauchy sequence wich doesn't converge.