What is wrong with this construction of the real numbers? Define a real number $r$ to be
$$r = x + \sum_{k = 0}^\infty \dfrac{a_k}{10^k}$$
where $x$ is a nonnegative integer and $a_k \in \{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$.   Denote the additive inverse of $r$ by $-r$.
 A: Note that even if you restrict the $a_{i}$ to come from the set $\{0,1,..,9\}$ you will still encounter issues. Consider $r_{1}=\sum^{\infty}_{k=1}\frac{a_{k}}{10^{k}}$ versus $r_{2}=1+\sum^{\infty}_{k=1}\frac{b_{k}}{10^{k}}$ where $a_{i}=9$ for all $i$ and $b_{i}=0$ for all $i$. These two expressions represent the same real number yet are defined with different $x$s and $\{a_{i}\}$s. To 'well define' the real numbers you will have to take steps (like the traditional 'equivalence classes of Chaucy sequences') to quotient out such coincidences.
A: If you confine $0\le a_k \le 9$ - nothing.
However, the definition is dependent on base 10, and various properties about limits/completeness need to be proved. I think Prof T Gowers has a note somewhere about how this works (couldn't find it immediately, to my shame).
The normal characterisations of the reals have been fine-tuned to the properties which seem to matter in further mathematical development. They can be shown to be equivalent (under appropriate conditions).
[Note Kaya's answer, which I know the aforementioned Prof Gowers has dealt with in his characteristically efficient fashion]
