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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!

  • 1
    Suppose $x$ were the real number $\pi$ which, as you know, has a decimal expansion that starts $3.14159265\dots$. Can you see any relation between this expansion and your question?2012-10-04

1 Answers 1