I have a problem with this exercise:
How many sequences of rational numbers converging to 1 are there?
I know that the number of all sequences of rational numbers is $\mathfrak{c}$. But here we count sequences converging to 1 only, so the total number is going to be less. But is it going to be $\mathfrak{c}$ still or maybe $\aleph _0$?
Best Answer
The number of sequence of rational numbers converging to $1$ is not countable. Suppose you get all squences by $(a_n^{(1)})_{n\in \mathbb{N}}, (a_n^{(2)})_{n\in \mathbb{N}}, (a_n^{(3)})_{n\in \mathbb{N}}, \dotsc$ Define a sequence $(b_n)_{n\in \mathbb{N}}$ by $$b_{k}:=\begin{cases}1 & a_k^{(k)}\neq 1\\ 1+\frac{1}{n}& a_k^{(k)}= 1 \end{cases}$$ Then $\lim_{n\to\infty}b_n=1$ but $(b_n)_{n\in \mathbb{N}}\neq (a_n^{(i)})_{n\in \mathbb{N}} $ for all $i\in\mathbb{N}$.