countable or uncountable sets NBHM 2016

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.