Proving $l^{\infty}$ not separable.

Question

I have proved the fact that $l^\infty$is not separable.I just want to verify whether my proof is correct.Suppose $A$ is a countable set in $l^\infty$.We have to show $A$ is not dense.Suppose $A=\{r_1,r_2,....\}$,then we construct a sequence $\xi$ as follows:

$\xi_i=1$ if $|r_{ii}-1|\nless 1/2$ and $\xi_i=0$ if $|r_{ii}-1|<1/2$,then $\xi\in l^\infty$ and $r_n \notin B(\xi,{1\over 2}) ,\forall n\in \mathbb N$.So,$B(\xi,{1\over 2})\cap A=\phi$,so $A$ is not dense.Is the proof alright,or some modifications can be made to make it much easier and shorter?

Answer 1

The proof is fine, in essence a diagonal argument to show that any countable subset cannot be dense in $\ell^\infty$.