Prove that $C^{\infty}_b(\mathbb{R}^{n})$ is dense in $C_{b}(\mathbb{R}^{n})$ using generic functions.

If $f$ is uniformly continuous then let $\phi \in C^\infty_c, \int \phi=1,\phi_k(x) =k^n \phi(kx)$ you'll have $f \ast \phi_k \to f$ uniformly ($*$ for convolution).

If $f$ is only continuous then split it in $f=\sum_{m\in \mathbb{Z}^n} f_m$ where $ f_m = f \prod_{j=1}^n (1-|x_j-m_j|) 1_{x_j-m_j\in [-1,1]}$ and look at $\sum_m f_m \ast \phi_{e_{m,k}}$ where $e_{m,k}$ is large enough such that $ \sup_{|x-y| < 1/e_{m,k}} |f_m(x)-f_m(y)| < 1/k$