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Let $\text{S}$ be the set of rational numbers with the following properties:

- $\frac{1}{2}\in \text{S}$
- If $x\in \text{S}$ then both $\frac{1}{x+1}\in \text{S}$ and $\frac{x}{x+1}\in \text{S}$

Which of the following is true?

- $\text{S}$ contains all rational numbers in the interval $0 < x < 1$.
- $\text{S}$ contains all rational numbers in the interval $-1 < x < 1$.
- $\text{S}$ contains all rational numbers in the interval $-1 < x < 0$.
- $\text{S}$ contains all rational numbers in the interval $1 < x <\infty$.