Is $(c_{00}, \|.\|_\infty)$ is not a Banach space?
if $c_{00} = \{x = (x_1, x_2, . . .) : x_n = 0$ for all but finitely many $n\}$ with $\|.\|_\infty$.
If I let the following sequence $v_n = (1,\frac{1}{2},\frac{1}{4},\dots,\frac{1}{2^n},0,0,\dots)$, then $v_n\in l^\infty$ and $v_n\in c_{00} $. So we have that $v_n \to v$ where ;
\begin{align} & v_n = (1,\frac{1}{2},\frac{1}{4},\dots,\frac{1}{2^n},0,0,\dots) \\ & v = (1,\frac{1}{2},\frac{1}{4},\dots,\frac{1}{2^n},\frac{1}{2^{n+1}},\dots) \ \ \ \ \text{(no zeros)} , \ \ \ \ \sup|\frac{1}{2^n}|<\infty \implies v\in l^\infty\\ \end{align}
Hence, $v\in l^\infty$ but $v\notin c_{00} $ as we can see above. So $c_{00} $ is not a closed subset of $l^\infty$, thus it is incomplete.