 Solution

Map each bounded sequence $a=(a_n)$ to the continuous function that takes value $a_n$ at the integer $n\ge 1$ and is affine on every interval $[n,n+1]$. Convexity of absolute value gives $|F_a(t)|\le {\sup }_n|a_n|$, and evaluation at integers gives equality of suprema. Thus $a\mapsto F_a$ is an isometric embedding of the l-infinity sequence space into $C^b([1,\infty ))$.

The former is not separable by the allowed fact. If the latter were separable, the subspace result just proved would make its isometric copy of ${\ell }_{\infty }$ separable, a contradiction. Hence