Let $S$ be the set of arithmetic sequences $(a_n)_n$ in $\mathbb{Z}$, i.e. there exists $d\in\mathbb{Z}$ such that $\forall n\in\mathbb{N}: a_{n+1} -a_n=d$.
What is the cardinality of $S$?
I thought about it as all the functions from $\mathbb{N}\rightarrow\mathbb{Z}$, so that gives ${\aleph _0}^ {\aleph _ 0} $.
The answer was the same but the conclusion was due to the face that you can choose $a_n\in\mathbb{Z}$ and $d\in\mathbb{Z}$.
Is my conclusion is right?
No, not quite.
Arithmetic sequences are determined by the difference $d$ of two subsequent terms and the initial value $a_0$. That is, there is a bijection $$S\longrightarrow\mathbb{Z}\times\mathbb{Z}$$ $$(a_n)_n\longmapsto (d,a_0)$$ The inverse map is given by $a_n=a_0+n\cdot d$.
Therefore the cardinality is
$$|S|=|\mathbb{Z}\times\mathbb{Z}|=|\mathbb{Z}|=\aleph_0$$
This is strictly smaller than $\aleph_0^{\aleph_0}=2^{\aleph_0}=|\mathbb{R}|$.