Countable subet

  • #1
So, the task is to prove: Every infinite set has a infinite countable subset.


2. A set [tex]X[/tex] is countable if there exists a bijection [tex]\phi: \mathbb{N}\rightarrow X[/tex]


3. So here's what I have:

Let [tex]A[/tex] be an infinite set, and pick some [tex]a_{1}\in A[/tex]. Define [tex]S_{n}=\left\{a_{i} [/tex], [tex]i\in \mathbb{N} \left| 1\leq i \leq n \right\} [/tex].
Pick [tex]a_{n}\in (A - S_{n-1})[/tex] for each [tex]n \in \mathbb{N}[/tex], [tex]n>1[/tex].

Let [tex]X=\left\{a_{n}|n \in \mathbb{N}\right\}[/tex], and let [tex]\phi: \mathbb{N}\rightarrow X[/tex] by [tex]\phi(x)=a_{x}[/tex].

Then I can show that [tex]\phi[/tex] is a bijection, and thus I am done. Is this good? I'm sure there has to be a better way to do this.
 
Last edited:

Answers and Replies

  • #2
I think it's alright. It's essentially what I suggested, but my answer has disappeared.. maybe it was too explicit..?
 
  • #3
Dick
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I think it's alright. It's essentially what I suggested, but my answer has disappeared.. maybe it was too explicit..?
Your post didn't disappear. It's on the other thread called 'countable sbset'. Don't repost the same question Doom of Doom.
 
  • #5
I am sorry. My internet crashed right when i submitted the post, then when I went back later I didn't see my original post, so I made a new one.
 

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