Chapter 8.3, Problem 39E

The meaning of the decimal representation of a number 0.d,d₂d₃… (where the digit d_i is one of the numbers 0, 1, 2, ... , 9) is that

$0.d,d_2{d}_3\dots =\frac{d_1}{10}+\frac{d_2}{{10}^2}+\frac{d_3}{{10}^3}+\frac{d_4}{{10}^4}+\dots$

Show that this series always converges.