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Question:

Let $f$: $\mathbb{N} \rightarrow \mathbb{Z}$ be a function that is eventually zero. i.e: There exists some $N \in \mathbb{N}$ s.t $f(n)=0$ for all $n \geq \mathbb{N}$. Prove that the set of such functions is countable.

Proof:

Define $g$: $\mathbb{N} \rightarrow \mathbb{Z}$ s.t $$ g(i) = \left\{ \begin{array}{ll} f(i) & \quad i \in \{0,1…N-1\} \\ 0 & \quad , otherwise \end{array} \right. $$

Let $B_n=\{h|h: \{0,1…N-1\}\rightarrow \mathbb{Z}$}. Clearly, $B_n$ is equal to $N$ copies of $\mathbb{Z}$. Hence, it's countable.

Let $G=\{f|f: \mathbb{N}\rightarrow \mathbb{Z}\}$ is eventually zero}. So, $G = \cup_{N=1}^{\infty} B_n$. Countable union of countable sets is countable. So is $G$.

Is my approach correct?

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The basic idea is good, but you rather want to take the union of $G_N$'s where $$G_N:=\{f:\Bbb N\to\Bbb Z:f(n)=0\text{ if } n>N\}$$

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Your proof is correct. Here are some points of feedback.

  • Your definition of $g$ depends on a specific $f$ and is actually just the same as that $f$. It does not really do anything in your proof, so just leave it out.
  • It is not quite true that $G = \bigcup_{N=1}^\infty B_n$, because the maps in $B_n$ do not have domain $\mathbb{N}$. They can however easily be made into maps with domain $\mathbb{N}$ (which is clearly what you mean), you should say this (and how).
  • A few small issues with notation, it is a bit sloppy. There is a $\}$ too many in the definition of $G$. The set $B_n$ depends on $n$ not on $N$ in its notation. That kind of thing. Not really a problem here, but in bigger proofs it will get confusing.
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  • $\begingroup$ For the first point, Think I should've used some other function, say $f'(i)$ instead of $f(i)$. Second, I think I should've defined a set $T_N$={$g: f' \in B_N$}? Then, take unions of all such $T_N$'s , which is equal to $G$? $\endgroup$ – SL_MathGuy Jan 1 at 14:39
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    $\begingroup$ @SL_MathGuy That sounds about right. It would be good to make explicit that $g$ depends on $f'$ (e.g. denote $g_{f'}$). Because what you are saying is that $g$ is the extension of $f'$ to $\mathbb{N}$ by setting it to $0$ everywhere $f'$ is not defined. $\endgroup$ – Mark Kamsma Jan 1 at 14:46
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I think there is a problem defining the map $g$...What you call $f$ is actually a set of functions....And also you do not use $g$ after you define it...

Some hint, How many maps from $\Bbb{N}$ to $\Bbb{Z}$ can you construct that they are zero after $n=0$ or after $n=1$,.....?

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