TITLE: How many sequences of rational numbers converging to 1 are there?
QUESTION [19 upvotes]: 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 $\aleph _0$?

REPLY [4 votes]: We have that non-repeating (injective) sequences of elements in $\{\,1+1/n:n\in\mathbb{N}\,\}$ form a continuum, and all of them have limit $1$, so our set is at least a continuum. Since also all rational sequences form a continuum, our set is also at most a continuum.