Proving Union of Countable Sets is Countable

[IntroAnalysis]

Homework Statement

If S is a countable set and {Ax}(s element S) is an indexed family of countable sets, then
U(s element S) As is a countable set.

Homework Equations

The Attempt at a Solution

S is countable means it is finite or countable infinite ( S equivalent to J set of positive integers).

As is countable also finite or countable infinite. Can I assume it is a subset of S?

If x element Union(s element S) As

I don't know how to prove the union of As is 1-1 and onto J? Can someone suggest how to start?

 

[mighty2000]

Hello, thank you for your post. To prove that the union of As is countable, we can use a proof by contradiction. Assume that the union of As is uncountable. This means that there exists a bijection f: J -> Union(s element S) As, where J is the set of positive integers and As is the indexed family of countable sets.

Since f is a bijection, every element in Union(s element S) As is mapped to a unique element in J. However, since As is a subset of S, this means that every element in As is also mapped to a unique element in J. But this contradicts the fact that S is countable, as there cannot exist a bijection between a countable set and a subset of itself.

Therefore, the union of As must be countable.