(a) Give Bernstein's probabilistic proof of Weierstrass's theorem.

(b) Are the following statements true or false? Justify your answer in each case.

(i) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there exists a sequence of polynomials $P_{n}$ converging pointwise to $f$ on $\mathbb{R}$.

(ii) If $f: \mathbb{R} \rightarrow \mathbb{R}$ is continuous, then there exists a sequence of polynomials $P_{n}$ converging uniformly to $f$ on $\mathbb{R}$.

(iii) If $f:(0,1] \rightarrow \mathbb{R}$ is continuous and bounded, then there exists a sequence of polynomials $P_{n}$ converging uniformly to $f$ on $(0,1]$.

(iv) If $f:[0,1] \rightarrow \mathbb{R}$ is continuous and $x_{1}, x_{2}, \ldots, x_{m}$ are distinct points in $[0,1]$, then there exists a sequence of polynomials $P_{n}$ with $P_{n}\left(x_{j}\right)=f\left(x_{j}\right)$, for $j=1, \ldots, m$, converging uniformly to $f$ on $[0,1]$.

(v) If $f:[0,1] \rightarrow \mathbb{R}$ is $m$ times continuously differentiable, then there exists a sequence of polynomials $P_{n}$ such that $P_{n}^{(r)} \rightarrow f^{(r)}$ uniformly on $[0,1]$ for each $r=0, \ldots, m$.