Show that $\left(a_n\right)_{n=1}^{\infty }$ is a Cauchy sequence of rational numbers.

Question

let $\left(d_k\right)_{k=1}^{\infty }\:$ be a sequence of numbers in the set $\left(1,2,3,4,5,6,7,8,9\right)\:$ Assume $a_n=\sum _{k=1}^n\left(\frac{d_k}{10^k}\right)$ for $n\in \mathbb{Z}^+$. Show that $\left(a_n\right)_{n=1}^{\infty }$ is a Cauchy sequence of rational numbers.

Answer 1

For $n \in \mathbb N$ and $m \gt 0$ you have

$$\vert a_{n+m} - a_n \vert= \sum_{k=n+1}^{n+m} \frac{d_k}{10^k} \le \sum_{k=n+1}^\infty \frac{9}{10^k} = \frac{1}{10^n}$$

Proving that $\{a_n\}$ is Cauchy. It is also obviously a sequence of rational numbers.