Proving that repeating decimals can be rewritten as fractions without using infinite series I'm being asked to prove that all repeating decimals can be written as fractions. The catch is that I'm not allowed to use infinite series, so that excludes most if not all methods I've seen so far.
I think that one way of doing this is letting $ x = .a_1a_2...a_k\overline{c_1c_2...c_d}  $ and showing that x can be rewritten as $ x = \frac{(10^d-1)*a_1a_2...a_k+c_1c_2...c_d}{(10^d-1)*10^k} $.
So does this sound valid? If so, how would I proceed with writing a meaningful proof provided these restraints?
 A: Perhaps the person setting the question regards this calculation as "avoiding infinite series": If
$$
x = 0.a_{1}a_{2}\dots a_{k}\overline{c_{1}c_{2}\dots c_{d}},
$$
then
\begin{alignat*}{2}
10^{d} 10^{k}x = \,
  &&a_{1}a_{2}\dots a_{k}c_{1}c_{2}\dots c_{d}
   &.\overline{c_{1}c_{2}\dots c_{d}}, \\
10^{k}x = \,
  &&a_{1}a_{2}\dots a_{k}
   &.\overline{c_{1}c_{2}\dots c_{d}}.
\end{alignat*}
Subtracting the second from the first,
$$
10^{k}(10^{d} - 1)x = a_{1}a_{2}\dots a_{k}(10^{d} - 1) + c_{1}c_{2}\dots c_{d}
$$
is an integer, so $x$ is indeed given by your formula, and in particular is rational.
A: Simplyfy your task. 
The nonrepeating part is clearly rational as: $0.a_1a_2\ldots a_k = \frac{a_1a_2\ldots a_k}{10^k}$
The repeating part is then: $10^k \times\frac{c_1c_2\ldots c_d}{10^d}\times 0.\overline{00\dots 01}$. 
The sum of two rational numbers is rational, as is the product.  
So you merely have to show that a repetition of a string of $(d-1)$ zeros and a one is rational.
$$0.\overline{1}=\frac 19, 0.\overline{01}=\frac 1{99}, \text{ et cetera.}$$
Show that every such repeating number is the singular solution that satisfies a linear equation with integer coefficients.
IE: $x = 0.\overline{\underbrace{0\ldots\ldots\ldots0}_{(d-1) \text{repeats}} 1} \\ 10^d x = 1.\overline{\underbrace{0\ldots\ldots\ldots0}_{(d-1) \text{repeats}} 1} \\ 10^d x - x = 1 \\ x= \frac{1}{10^d-1} \\ \therefore x \in \mathbb{Q}$
$\mathbb{QED}$ All numbers with a non-terminating representation that eventually becomes periodic (aka repeating, or recurring, decimals) are rational numbers.
