I was wondering: let be $r_n$ a sequence of numbers randomly (i.d.d.) chosen from $[0, 1]$ so $r_n \in [0, 1]-\{r_1, r_2, ... ,r_{n-1}\}$. And any r is a real number. Is we call $s(r_n)$ the space of all subsequences of $r_n$... What is the probability of pick a convergent subsequence from $s(r_n)$?
My intuition says that the probability is $0$, but I can't imagine any demonstration. Can you help me, please?
Many thanks in advance!
EDIT: Does it change in any way the outcome of this problem if we choose all $r_n$ to be rational numbers?
It is indeed 0. This can be shown in the following way:
For the chosen sequence to be convergent, there must be a point in the sequence where all points are within an $\epsilon$ boundary of the limit of the sequence. As such, let's take $\epsilon=\frac14$. In this case, the probability of each element of the sequence after this point being within $\frac14$ of the limit is at most $\frac12$. So, as we pick more and more terms of the sequence, the probability of all of them being within some range goes down to 0