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Consider the set of fractions listed in Figure 7.3 on page 165. Let's call this set $F$. Notice that $\mathbb{Q}\subseteq F$ (we're not reducing the fractions in $F$). If we can show that $F$ is countable, then we're done (by Theorem 7.3.4). The general strategy here is to construct a function $f: \mathbb{N}\to F$ that is onto and then appeal to Theorem 7.3.5. Doing this is rather tricky and probably best to do by drawing a picture. Try to find a path through the grid in Figure 7.3 that hits every fraction exactly once. Define $f: \mathbb{N}\to F$ via $f(n)$ is the fraction that you hit on the $n\text{th}$ step in your path.

If you can figure out the path by drawing a picture, I can help you scan it and post it here.

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