I came across a problem I would like to ask you about:
Let $x$ be a real number. Iwant to show that $\exists b \in Z$ and integers $ b_1, b_2,b_3, \ldots\in \left\{{0,1,\ldots,9}\right\}$ so that the sequence
$s_n := b + \sum\limits_{k=1}^n 10^{-k}\cdot b_k$
converges to the real number $x$.
I can imagine that the sequence looks like ($b=1$ f.ex.) $1,1+\frac{b_1}{10}, 1+\frac{b_1}{10}+\frac{b_2}{100}, \ldots$
But I don't know how to approach this. Is the goal to show that it converges, or that the limit is a real number?
Any help is greatly appreciated!