Explanation:

A set is said to be countable if its elements can be put into a one-to-one correspondence with the natural numbers (1, 2, 3, ...). An uncountable set, on the other hand, cannot be put into a one-to-one correspondence with the natural numbers.

To understand why the correct answer is option 'A' (every infinite set has a countable subset), let's consider the following points:

1. Countable Sets:
- Countable sets are those sets that can be enumerated using the natural numbers.
- Examples of countable sets include the set of natural numbers, integers, and rational numbers.
- Countable sets can be finite or infinite.

2. Uncountable Sets:
- Uncountable sets are those sets that cannot be enumerated using the natural numbers.
- Examples of uncountable sets include the set of real numbers, irrational numbers, and the power set of natural numbers.
- Uncountable sets are always infinite.

3. Existence of Countable Subsets:
- For any infinite set, it is always possible to find a countable subset within it.
- This is because we can start enumerating the elements of the set using the natural numbers, and we will eventually cover all the elements of the infinite set.
- Even if the set is uncountable, we can still find a countable subset within it.

4. Cantor's Diagonal Argument:
- Cantor's Diagonal Argument is a mathematical proof that shows that the set of real numbers is uncountable.
- This argument demonstrates that it is not possible to put the real numbers into a one-to-one correspondence with the natural numbers.
- However, this does not imply that there are no countable subsets within the set of real numbers.

Conclusion:
- Every infinite set, whether countable or uncountable, has a countable subset.
- Therefore, the correct answer to the question is option 'A' (every infinite set has a countable subset).