countable or uncountable sets NBHM 2016

Question

1. The set of all algebraic numbers
2. the set of all strictly increasing infinite sequences of positive integers
3. the set of all infinite sequences of integers which are in arithmetic progression .

I know 1 is definitely countable set. and i am not sure about 2 and 3 but it seems uncountable as they are infinite sequences. Thanks in advance

Answer 1

2 is uncountable as those sequences effectivly just represent the infinite subsets of $\Bbb N$. Since the set of all subsets in uncaountable and the set of finite subsets is countable, the set of infinite subsets of $\Bbb N$ is uncountable.

3 is countable because each such sequence can be encoded using just two integers (first term and difference, say) and $\Bbb N\times \Bbb N$ is countable.

Answer 2

2. Let $C=\{0,1\}^{\mathbb{N}}$ then it is known that $C$ is not coutable. The function $f: C\to \mbox{the set of all strictly increasing infinite sequences of positive integers}$, $f((a_n))=(a_n +2n)$ is one to one hence the set of all strictly increasing infinite sequences of positive integers is not countable.