I try to show that the space $c_{00}=\{(x_n):x_n=0 \text{ all but finitely many }n\}$ is not complete with respect to the norm $\|x\|_\infty=\max |x_n|$.
My attempt:
Let $(z_n)=\left(1,\frac{1}{2},\dots,\frac{1}{n},0,0,\dots\right)$ be a sequence. Clearly $(z_n)\in c_{00}$. We have convergence $(z_n)\to (z_\infty)$ with $(z_\infty)\in c_{0}$ but $(z_\infty) \not\in c_{00}$. So we have only to show that $(z_n)$ is a Cauchy sequence with respect to $\|\|_\infty$.
Now I am confused because of the $\|\cdot\|_\infty$ norm. Do I just subtract the entries of the sequences and then take the maximum entry? Meaning for $m>n$ I have $$z_n-z_m=\left(0, \dots,0,\frac{1}{n+1},\dots,\frac{1}{m},0,0,\dots\right)$$ $$\|z_n-z_m\|_\infty=\frac{1}{n+1}\to 0 \text{ as } n\to \infty $$
Thank you for help
What you did is fine. Now, given $\varepsilon>0$, take $N\in\mathbb N$ such that $\frac1N\leqslant\varepsilon$. Then$$m>n\geqslant N\implies\lVert z_n-z_m\rVert=\frac1{n+1}\leqslant\frac1{N+1}<\frac1N\leqslant\varepsilon.$$