Suppose I have a positive number $d \in \mathbb{R}$ and a sequence of numbers $a_n \in [0,d]$ for $n \in \mathbb{N}$ with the following properties $$ \sum_{i=1}^{\infty} a_i^r \in \mathbb{Z} $$ for all $r \in \mathbb{N}$ and $$ \sum_{i=1}^{\infty} a_i \leq d \ . $$

Does it follow that only finitely many of the $a_i$ are non-zero?

Note that it does not follow that $a_i \in \mathbb{Z}$ as the sequence $a_1 = 1 + \sqrt{2}$, $a_2 = 1 - \sqrt{2}$, $a_k = 0$ for $k > 2$ with $d=3$ shows. This sequence also shows that the power sums can be unbounded.

Suppose I have a positive number $d \in \mathbb{R}$ and a sequence of numbers $a_n \in [0,d]$ for $n \in \mathbb{N}$ with the following properties $$ \sum_{i=1}^{\infty} a_i^r \in \mathbb{Z} $$ for all $r \in \mathbb{N}$ and $$ \sum_{i=1}^{\infty} a_i \leq d \ . $$

Does it follow that only finitely many of the $a_i$ are non-zero?

Note that it does not follow that $a_i \in \mathbb{Z}$ as the sequence $a_1 = 2 + \sqrt{2}$, $a_2 = 2 - \sqrt{2}$, $a_k = 0$ for $k > 2$ with $d=4$ shows. This sequence also shows that the power sums can be unbounded.