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 $\aleph _0$?

I believe there is a continuum number of such sequences. For ease of writing, let's try to equivalently count the number of sequences converging to $0$. First note that there are countably many rational numbers in the interval $[-a,a]$. Now set for every sequence we will consider $x_i \in [\frac{-1}{i},\frac{1}{i}]$, so that each sequence of $x_i$'s will converge to $0$. The number of such sequences is equivalent to the number of functions from $\mathbb{N}$ to itself, which is provably uncountable.
