1. ## Countable Union proof

The countable union of countable sets is countable, i.e., if An is a countable set for each n in the Naturals then the union from n=1 to infinity of An is countable.

Any help with this proof would be appreciated!

2. Let (n,m) denote the m-th element of the n-th countable set.

(1,1), / (1,2),(2,1), / (1,3),(2,2),(3,1), / ...

(The slashes are just to emphasize the grouping.) In words, you're taking an element from the first set, then one each from the first two sets, then one each from the first three sets, and so forth. It's easy to see that for any particular element in any particular set, you will pick it at some finite step.