Exercise 1.5.1

Question

Finish the following proof for Theorem 1.5.7.
  Assume $B$ is a countable set. Thus, there exists $f: \mathbf{N} \rightarrow B$, which is $1-1$ and onto. Let $A \subseteq B$ be an infinite subset of $B$. We must show that $A$ is countable.

Let $n_{1}=\min \{n \in \mathbf{N}: f(n) \in A\}$.
  As a start to a definition of $g: \mathbf{N} \rightarrow A$ set $g(1)=f\left(n_{1}\right)$.
  Show how to inductively continue this process to produce a 1-1 function $g$ from $\mathbf{N}$ onto $A$.

Ulisse Mini · Accepted Answer

Let $n_k = \min\{n \in \mathbf N \mid f(n) \in A, n 
otin \{n_1, n_2, \dots, n_{k-1}\}\}$
  and $g(k) = f(n_k)$. since $g : \mathbf N 	o A$ is 1-1 and onto, $A$ is countable.

Exercise 1.5.1

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Exercise 1.5.1

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