Show the following.(i) If $X$ is a random variable with $|X| \leq 1$ a.s., then there is a random variable $Y$ with values in $\{-1,+1\}$ and with $\mathbf{E}[Y \mid X]=X$
(ii) For $X$ as in (i) with $\mathbf{E}[X]=0$, infer that (using Jensen's inequality)
$$
\mathbf{E}\left[e^{\lambda X}\right] \leq \cosh (\lambda) \leq e^{\lambda^{2} / 2} \quad \text { for all } \lambda \in \mathbb{R}
$$
(iii) If $\left(M_{n}\right)_{n \in \mathbb{N}_{0}}$ is a martingale with $M_{0}=0$ and if there is a sequence $\left(c_{k}\right)_{k \in \mathbb{N}}$ of nonnegative numbers with $\left|M_{n}-M_{n-1}\right| \leq c_{n}$ a.s. for all $n \in \mathbb{N}$, then
$$
\mathbf{E}\left[e^{\lambda M_{n}}\right] \leq \exp \left(\frac{1}{2} \lambda^{2} \sum_{k=1}^{n} c_{k}^{2}\right)
$$
(iv) Under the assumptions of (iii), Azuma's inequality holds:
$$
\mathbf{P}\left[\left|M_{n}\right| \geq \lambda\right] \leq 2 \exp \left(-\frac{\lambda^{2}}{2 \sum_{k=1}^{n} c_{k}^{2}}\right) \quad \text { for all } \lambda \geq 0
$$