Can every real number be written in decimal expansion?

I mean, can every real number $a$ be expressed as follows:
$$\text{For }\, a \in \mathbb {R}^{+},\quad a=p+\frac{n_{1}}{10}+\frac{n_{2}}{10^{2}}+\cdots$$
$$\text{where }\, n_{i} \in \mathbb N \quad \text{and} \quad 0\le n_{i}<10, \;\; p \in \mathbb Z$$

  • $\begingroup$ If you have changed $n$ for $p \in \mathbb{Z}$, you don't need restrict $a$ to $\mathbb{R}^+$. $\endgroup$ – Alex Silva Jan 2 '15 at 13:46

Yes, every real number can be written as a decimal expansion. For a proof, see Rudin Principles of Mathematical Analysis (McGraw Hill 1976) p. 11.

Moreover, for every real number $r$ not of the form $\frac n{10^k} (n,k\in \Bbb Z)$, the decimal expansion is unique.


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