Solution
= Solution
The finite truncations of $x=(1,1/2,1/3,\ldots)$ lie in $c_{00}$ and converge in the <supremum norm> to $x\notin c_{00}$. Thus \b[$c_{00}$ is not closed.] Every element of $c_0$ is the norm limit of its finite truncations, while $c_0$ is closed and contains $c_{00}$. Hence \b[$\boxed{\overline{c_{00}}=c_0}$.]