Introduction
A countable set is a set that has the same cardinality (number of elements) as the set of natural numbers (1, 2, 3, ...). In other words, a countable set can be put into a one-to-one correspondence with the set of natural numbers. Countable sets can be finite or infinite, but they are always considered to be "small" in terms of cardinality.

Countable Union of Countable Sets
A countable union of countable sets is a collection of countable sets that are combined together into a single set. This union is also countable, meaning that it has the same cardinality as the set of natural numbers.

Proof by Example
Let's consider a specific example to understand this concept better. Suppose we have a countable collection of countable sets:

A1 = {1, 2, 3, ...}
A2 = {4, 5, 6, ...}
A3 = {7, 8, 9, ...}
...

Each countable set Ai can be put into a one-to-one correspondence with the set of natural numbers. Now, if we combine all these countable sets together into a single set, we get:

A = A1 ∪ A2 ∪ A3 ∪ ...

This union A is also countable because we can create a one-to-one correspondence between the elements of A and the set of natural numbers. We can do this by listing the elements of A in a specific order, such as:

A = {1, 2, 3, 4, 5, 6, 7, 8, 9, ...}

Uncountable Sets
An uncountable set, on the other hand, has a cardinality that is greater than that of the set of natural numbers. In other words, it cannot be put into a one-to-one correspondence with the set of natural numbers.

Proof by Contradiction
To show that a countable union of countable sets is not uncountable, we can use a proof by contradiction. Suppose we assume that a countable union of countable sets is uncountable. This means that there exists an uncountable set B that is the countable union of countable sets:

B = A1 ∪ A2 ∪ A3 ∪ ...

However, we have already shown that a countable union of countable sets is countable. This contradicts our assumption that B is uncountable. Therefore, our assumption must be false, and a countable union of countable sets is not uncountable.

Conclusion
In conclusion, a countable union of countable sets is countable, not uncountable. This can be proven by showing that the union can be put into a one-to-one correspondence with the set of natural numbers.