Show that $(C_b (0,1],||\cdot||_{\infty})$ is not separable I'm not sure what I'm suppose to do for this question. Any help would be greatly appreciated. I know that there is a similar question like this for $[0,1)$ instead. But is there anyway for me to show this without having to prove the isomorphism between $[0,1)$ and $(0,1]$?

Hint: Think of triangles of height $1$ over the intervals $[1/(n+1),1/n], n = 1,2,\dots.$ 
