(a) Let $\left(x_{n}\right)_{n=1}^{\infty}$ be a non-negative and decreasing sequence of real numbers. Prove that $\sum_{n=1}^{\infty} x_{n}$ converges if and only if $\sum_{k=0}^{\infty} 2^{k} x_{2^{k}}$ converges.

(b) For $s \in \mathbb{R}$, prove that $\sum_{n=1}^{\infty} n^{-s}$ converges if and only if $s>1$.

(c) For any $k \in \mathbb{N}$, prove that

$$\lim _{n \rightarrow \infty} 2^{-n} n^{k}=0$$

(d) The sequence $\left(a_{n}\right)_{n=0}^{\infty}$ is defined by $a_{0}=1$ and $a_{n+1}=2^{a_{n}}$ for $n \geqslant 0$. For any $k \in \mathbb{N}$, prove that

$$\lim _{n \rightarrow \infty} \frac{2^{n^{k}}}{a_{n}}=0$$