The Clue Is in the Question

This problem is in two parts. The first part provides some building blocks which will help you to solve the final challenge. These can be attempted in any order. This problem can also test your powers of conjecture and discovery: As you start from one of the mini-challenges, how many of the other related mini-challenges will you invent for yourself?

This challenge involves building up a set $F$ of fractions using a starting fraction and two operations which you use to generate new fractions from any member of $F$.

Rule 1: $F$ contains the fraction $\frac{1}{2}$.

Rule 2: If $\frac{p}{q}$ is in $F$ then $\frac{p}{p+q}$ is also in $F$.

Rule 3: If $\frac{p}{q}$ is in $F$ then $\frac{q}{p+q}$ is also in $F$.


Choose a mini-challenge from below to get started. There is a lot to think about in each of these mini-challenges, so as you think about them, continually ask yourself: Do I have any other thoughts? Do any other questions arise for me? Make a note of these, as they might help when you consider other parts of the problem.

Mini-challenge A	Which of these fractions can I reach? $$ \frac{1}{2}\,, \frac{1}{7}\,, \frac{2}{7}\,, \frac{5}{9}\,, \frac{11}{13}\,, \frac{17}{16}\,, \frac{19}{8}\,, \frac{2}{1}\,, $$
Mini-challenge B	What is the biggest/smallest fraction you can make? What is the biggest numerator/denominator you can make?
Mini-challenge C	Is it true that the numerators never decrease?
Mini-challenge D	Can I make a fraction for which the numerator and denominator have a common factor?
Mini-challenge E	Can I make a 'closed loop': a sequence of transformations which end up back at the starting point?
Mini-challenge F	Can you make sense of the process of working backwards from various fractions?
FINAL CHALLENGE	Show that every rational number between $0$ and $1$ is in $F$