Are the following sets countable or uncountable?

Question

1) $\bigcup_{n \in \mathbb{N}}^{\infty} A_n$ , if $A_n$ countable.

2) $\mathbb{R}$\ $\mathbb{Q}$

3) $\bigcup_{n \in \mathbb{N}}$ { ($a_1$,...,$a_n$) | $a_1$, ..., $a_n$ $\in$ $\mathbb{Z}$ }

4) {$(a_n)_{n \in \mathbb{N}}$ | $(a_n)_{n \in \mathbb{N}}$ null sequence in $\mathbb{R}$ }

I've already proved 1) and 2) ( 2) is uncountable ) . But I failed to prove 3) and 4) . I think that 3) and 4) are uncountable?

Answer 1

For 3, you should already have a proof that the set of ordered pairs of integers is countable. You can extend that to the fact that the set of ordered $n-$tuples is countable. For triples, you have bijections $(a,b,c)\leftrightarrow ((a,b),c) \leftrightarrow(d,c)$ This approach extends to any $n$. Then use part 1 to form the union of all the tuples of any finite length.