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## Homework Statement

I am trying to show that a family F {A[tex]\subseteq[/tex][tex]N[/tex]: A is finite} Show that F is countable.

I found a proof in a book but I don't understand it. Could you help me please?

## Homework Equations

## The Attempt at a Solution

If A is finite, then F is clearly also finite and the claim becomes trivially true. We therefore suppose that A is countable infinite.

To prove that F is countable it would be sufficient to show that the family of all finite subsets of "Naturals" is countable. For each n an element of "Naturals", let M

_{n}denote the family of all subsets N of "Naturals" satisfying N[tex]\subseteq[/tex] {1,2,...,n}.

M

_{n}is obviously finite.

If M is the family of all finite subsets of "Naturals", we see that it can be expressed as M= the union of all M

_{n}from n=1 to infinity. From the theorem that "union of countable sets is countable" we see though that M is countable and hence see that the family F is also countable.

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My first question is as follows. In the beginning, how do they know that if A is finite, then F is clearly also finite?