Why is $\ell^\infty(\mathbb{N})$ not separable?

My functional analysis textbook says

"The metric space $l^\infty$ is not separable."

The metric defined between two sequences $\{a_1,a_2,a_3\dots\}$ and $\{b_1,b_2,b_3,\dots\}$ is $\sup\limits_{i\in\Bbb{N}}|{a_i-b_i}|$.

How can this be? Isn't the set of sequences containing complex numbers with rational coefficients the required countable dense subset of $l^\infty$?

Thanks in advance!