(a) If $f:(0,1) \rightarrow \mathbb{R}$ is continuous, prove that there exists a sequence of polynomials $P_{n}$ such that $P_{n} \rightarrow f$ uniformly on compact subsets of $(0,1)$.

(b) If $f:(0,1) \rightarrow \mathbb{R}$ is continuous and bounded, prove that there exists a sequence of polynomials $Q_{n}$ such that $Q_{n}$ are uniformly bounded on $(0,1)$ and $Q_{n} \rightarrow f$ uniformly on compact subsets of $(0,1)$.