$C([0,\infty))$ is separable

Question

In many books (e.g. Brownian Motion and Stochastic Calculus by Karatzas and Shreve) the fact that $C([0,\infty)):=\{f\colon [0,\infty)\to\mathbb{R} \mid f \text{ continuous}\}$ endowed with the metric $$ d(f,g) = \sum_{k=1}^\infty 2^{-k}\min\{\sup_{t\in[0,k]}|f(t)-g(t)|,1\} $$ is a complete separable metric space, is left as an exercise for the reader. I am certainly struggling to show that the space is separable. I would be grateful for any reference or hint.