Let $X$ be an integrable random variable with $\mathbb{E}(X)=0$. Show that the characteristic function $\phi_{X}$ is differentiable with $\phi_{X}^{\prime}(0)=0$. [You may use without proof standard convergence results for integrals provided you state them clearly.]

Let $\left(X_{n}: n \in \mathbb{N}\right)$ be a sequence of independent random variables, all having the same distribution as $X$. Set $S_{n}=X_{1}+\cdots+X_{n}$. Show that $S_{n} / n \rightarrow 0$ in distribution. Deduce that $S_{n} / n \rightarrow 0$ in probability. [You may not use the Strong Law of Large Numbers.]