Suppose A1, A2, A3,... is an infinite sequence of countable sets. Recall that 2 i=1 Ai =

Chapter 7, Problem 38

(choose chapter or problem)

Suppose \(A_{1}, A_{2}, A_{3}\), . . . is an infinite sequence of countable sets. Recall that

\(\bigcup_{i=1}^{\infty} A_{i}=\left\{x \mid x \in A_{i}\right.\) for some positive integer i }.

Prove that \(\bigcup_{i=1}^{\infty} A_{i}\) is countable. (In other words, prove that a countably infinite union of countable sets is countable.)

Text Transcription;

A_1, A_2, A_3

bigcup_i=1^infty A_i ={x mid x in A_i

bigcup_i=1^infty A_i