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I am trying to understand a proof from Mathematical Analysis by Apostol for the following theorem:

The set of rationals $\mathbb{Q}$ is countable.

Here is the proof (I rewrote a few things):

Let $A_n$ be the set of positive rational numbers that have denominator $n$. Then the following is true: $$\mathbb{Q}^+=\bigcup_{i=1}^\infty A_i$$ Since each $A_i$ is countable, $\mathbb{Q}^+$ is countable. Similarly for $\mathbb{Q}^-$ and $\{0\}$. Then $\mathbb{Q}=\mathbb{Q}^-\cup\{0\}\cup\mathbb{Q}^+$ is countable.

For some reason, I cannot understand the bold part. How is that each $A_{i}$ is countable?

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  • $\begingroup$ Think about A_2 for concreteness. There is one element of A_2 for every positive integer that you put in the denominator (or zero, if you don't want to count non-reduced fractions, but it doesn't matter either way). Explicitly, the elements of A_2 are 1/2, 2/2, 3/2, 4/2, 5/2, and so on $\endgroup$ – Brennan Vincent Jan 7 '19 at 4:07
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Consider the map $f_i:A_i\rightarrow \mathbb{N}$ defined by $f({a\over i})=a$, it is injective. This implies that $A_i$ has a cardinality of a subset of $\mathbb{N}$ so it is countable.

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Rephrasing:

Fix $i$ , $i \in \mathbb{Z^+}$.

$A_i= ${$1/i,2/i,3/i,.......$}.

$f(\mathbb{Z^+}) \rightarrow A_i,$

$f(n)= n/i$, $i$ fixed, is a bijection.

$A_i$ is countable.

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