(a) Prove that every real number $\alpha \in(0,1]$ can be written in the form $\alpha=$ $\sum_{n=1}^{\infty} 2^{-b_{n}}$ where $\left(b_{n}\right)$ is a strictly increasing sequence of positive integers.

Are such expressions unique?

(b) Let $\theta \in \mathbb{R}$ be a root of $f(x)=\alpha_{d} x^{d}+\cdots+\alpha_{1} x+\alpha_{0}$, where $\alpha_{0}, \ldots, \alpha_{d} \in \mathbb{Z}$. Suppose that $f$ has no rational roots, except possibly $\theta$.

(i) Show that if $s, t \in \mathbb{R}$ then

$$|f(s)-f(t)| \leqslant A(\max \{|s|,|t|, 1\})^{d-1}|s-t| .$$

where $A$ is a constant depending only on $f$.

(ii) Deduce that if $p, q \in \mathbb{Z}$ with $q>0$ and $0<\left|\theta-\frac{p}{q}\right|<1$ then

$$\left|\theta-\frac{p}{q}\right| \geqslant \frac{1}{A}\left(\frac{1}{|\theta|+1}\right)^{d-1} \frac{1}{q^{d}} .$$

(c) Prove that $\alpha=\sum_{n=1}^{\infty} 2^{-n !}$ is transcendental.

(d) Let $\beta$ and $\gamma$ be transcendental numbers. What of the following statements are always true and which can be false? Briefly justify your answers.

(i) $\beta \gamma$ is transcendental.

(ii) $\beta^{n}$ is transcendental for every $n \in \mathbb{N}$.