The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-separability of $L^\infty(\mathbb R)$ (reductio ad absurdum: given a sequence $(\phi_n)_{n\in \mathbb N}$ in $L^\infty(\mathbb R)$, construct a function in  $L^\infty(\mathbb R)$ at distance 1 of that sequence).

Is it possible to avoid that type of proof and get the non-separability of $C_b(\mathbb R)$ by a more abstract (structural) fact?

The vector space $C_b(\mathbb R)$ of bounded continuous functions on $\mathbb R$ is non-separable: it is possible to produce a direct proof of this fact, mimicking the standard proof for the non-separability of $L^\infty(\mathbb R)$ (reductio ad absurdum: given a sequence $(\phi_n)_{n\in \mathbb N}$ in $L^\infty(\mathbb R)$, construct a function in  $L^\infty(\mathbb R)$ at distance 1 of that sequence).

Is it possible to avoid that type of proof and get the non-separability of $C_b(\mathbb R)$ by a more abstract (structural) fact?