How many sequences of rational numbers converging to 1 are there? 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 $\aleph _0$?

For each real number $x$ define
$$a_n(x):= 1+ \frac{ \lfloor xn \rfloor}{n^2}$$
Show that this is a one-to-one function from $\mathbb R$ to the set of sequences converging to $1$.
