Cardinality, Subsets, Sequences

[Csharp]

Define
F(A) = The set of all finite subsets of A
Seq(A) = The set of all finite sequences with elements from A

Let A be an infinite set (not necessarily countable).
I want to prove the following lines.

1. Card seq(A) \(\displaystyle \le Card(A^\omega) \)
2. Card A = Card seq(A) = Card F(A)

 

[Valkarie]

1. Since seq(A) is a subset of A^ω, its cardinality must be less than or equal to the cardinality of A^ω. To prove this, let S be the set of all finite sequences with elements from A. We can see that |S| ≤ |A^ω|. 2. Since A is an infinite set, both F(A) and Seq(A) have the same cardinality of A. This follows from the fact that any element in A can be either a finite subset of A or a finite sequence with elements from A. Thus, |A| = |F(A)| = |Seq(A)|.