I am in need of help with the following question.
(a) Let $(d_{i})_{i=1}^{\infty}$ be a sequence of integers with $0\leq d_{i}\leq 9$. Prove that the series $\sum_{i=1}^{\infty}(d_{i}\cdot10^{-i})$ converges
(b) If the sequence $(d_{i})_{i=1}^{\infty}$ is periodic, then show that $\sum_{i=1}^{\infty}(d_{i}\cdot10^{-i})$ is a rational number. Show this number.
For part (a) I am thinking intuitively that the numbers in the series will be bounded. If $0\leq d_{i} \leq 9$ then $0 \leq d_{i}\cdot10^{-i} \leq 1$. Should I be thinking about taking the limit?
For part (b) I am unsure where to begin.