Follow up to an answer: Why is $l^{\infty}$ not separable?

Question

In this question thread, user940 answered and said that, "Assume that $A⊂l^∞$ is countable..."

I want to understand:

1. why can we assume something like this? can assuming something like this make our result less general?

2. can we do a similar proof without assuming that $A$ is countable, and instead by considering any sequence $a_k \in A$ such that $||a_k-b||>$ some $\epsilon?$