Which of the following are countable?
- The set of all algebraic numbers
- The set of all strictly increasing infinite sequences of positive integers.
- The set of all infinite sequences of integers which are in arithmetic progression.
My try:
1.countable ;since algebraic numbers are roots of equation of a polynomial over $\Bbb Q$ and $\Bbb Q$ is countable so the set is countable.
2.cardinality of set of all functions from $\Bbb N\to \Bbb Z^+$ is $\Bbb N^{\Bbb N}=\mathbb c$.So uncountable.
3.Unabe to proceed.
Please check my solutions and how to proceed with $3$
For the last point, you fix the starting point, the common difference, and the number of terms. So the number of arithmetic progression has cardinality $\mathbb{N}^3$, which is countable. Others are correct.
Edit: As Noah correctly pointed out below, the OP asks for infinite sequences in arithmetic progression. Here, you just fix the starting point and the common difference, so the number of these ones is still countable. As a shortcut, you cay regard any infinite arithmetic progression as the extension of a finite one, and since the cardinality of $\mathbb{N}$ is the "smallest" infinite, the same conclusion follows.