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let $k$ be an integer, and consider the set $S$ of functions from $\mathbb{N}$ to $\mathbb{N}$ where $g(x) =0$ for all $x>k$. I want to show $S$ is countable.

This is what I have so far, but I am not quite sure if I am going about it correctly, or what to do next.

So our functions will look like this

$A_1=\{(1,a_1),(2,a_2),(3,a_3),(4,a_4),....,(k,a_k), (k+1,0), (k+2,0)....\}$

It seems from here I should be able to show that there are only countable ways of doing this, but I am not quite sure how to say that. I think the best way would to come up with a bijection from the naturals to $S$ but for the life of me I can't come up with one.

I would love some help. Thanks in advance.

tmpys
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  • With the exception of one, all the questions you tagged as [elementary-set-theory] were originally tagged with [set-theory] as well. Please stop using both tags at once, and if you have a doubt about which one, it's probably the elementary set theory that fits better. – Asaf Karagila Nov 06 '14 at 06:47
  • ahh I am sorry, I thought we wanted the maximum amount of tags... Wont happen again. – tmpys Nov 06 '14 at 06:50
  • Please search the site before posting questions. – Asaf Karagila Nov 06 '14 at 07:38
  • I actually don't need that to be proved, as we proved it in class. So the question I asked actually has nothing to do with that. As you know, because you answered my question, I needed a hint that would lead me to the needed injection. Let me point out that you answered my question, and in your answer you first told me that I was looking for the bijection from S to N^K, and after you told me this, I showed you the bijection and then we moved on. Thus you know that I never even received any information about N^k being countable, and I checked answered, so I was never going to need such info. – tmpys Nov 06 '14 at 08:24
  • I just wanted to point this out for when I report that you are obviously harassing me. Our entire conversation will show the order in which these events took place. Thank you for the help on the math related stuff. – tmpys Nov 06 '14 at 08:26

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