While proving that the space $l^p$ with $1 \le p < \infty$ is separable, the book is suggested the following subset [which is easy to prove denseness…] :
$M$ to be the set of all sequences $y$ of the form $y = (r_1, r_2, \dots, r_n, 0, 0, \dots)$ where $n$ is any positive integer and the $r_j$'s are rational.
How to prove that $M$ is countable? $\sum_{i=1}^{\infty} \mathbb{Q^i}$ 'looks' big enough to be like $2^{\mathbb{N}}$ than finitely many product of $\mathbb{Q}$ to be countable.
I am not much familiar with axiomatic set theory. Simple clear explanation would be much appreciated.
Best Answer
$M$ is countable, because
$$ \begin{align*} \left|\sum_{i\in\mathbb{N}}\mathbb{Q}^i\right| &\leq\sum_{i\in\mathbb{N}}|\mathbb{Q}|^i\\ &=\sum_{i\in\mathbb{N}}\aleph_0^i\\ &=\sum_{i\in\mathbb{N}}\aleph_0\\ &=\aleph_0^2=\aleph_0 \end{align*} $$
Or, a bit more concretely, fix a bijection $f\colon\mathbb{Q}\to\mathbb{N}$, $f(0)=0$ and inject $M\to\mathbb{N}$ by $2^{f(r_1)}\cdot 3^{f(r_2)}\cdot 5^{f(r_3)}\cdots$