Reference on separability of $c_0(X)$, where $X$ is a separable Banch space

Question

Let $(X,\|\cdot\|)$ be a separable Banach space. By $c_0(X)$ I mean the space $\{(x_n)_n\subset X:\, \|x_n\|\to0\}$. I think it os well known that $c_0(X)$ is a separable Banach space endowed with the norm $$\|(x_n)_n\|_\infty:=\sup_n\|x_n\|.$$ I need a reference on that, but i'm not able to find it.

Can anybody help me?

Answer 1

I suspect most introductory Banach space books will provide this as either an example or an exercise; as the comments say this is quite straightforward to prove. However, a nice reference is Fabian, Habala, Hajek, Montesinos and Zizler; Banach Space Theory, page $22$, proposition $1.42$.