# How to define/What is a probability that random sequence of real numbers is convergent?

Inspired by Are half of all numbers odd?, idea of asymptotic density in this answer for odd numbers in set of all positive integers as $$\lim_{n \to \infty} \frac{|\{k \leq n, k \text{ is odd}\}|}{n},$$ (which is $$1/2$$), which can be viewed as a probability that random integer is odd (if we limit to numbers up to finite boundary $$n$$ and then going $$n\to\infty$$). In same spirit, I was wondering what is a probability that random sequence $$\{a_k\}_{k=1}^\infty$$ of real numbers is convergent? How do we even choose probability distribution (as in above case, it should be in some sense uniform)? Intuitively, I would expect the probability to be $$0$$ since there seems to be vastly more divergent sequences than convergent, but I was thinking the same about transcendental numbers, so... Also, not sure if this will be any helpful, but cardinality of convergent sequences is continuum, as this shows: What is the cardinality of the set of all sequences in $$\mathbb{R}$$ converging to a real number? ...

I wanted to try similar approach as above with odd integers, to that I would need to have some sequence of finite sets of sequences $$A_k$$ and $$B_k$$, where $$A_k$$ converges to set of all convergent sequences and $$B_k$$ converges to set of all sequences (and I am not even sure in what metric space that would be...), and then compute $$\lim \frac{|A_k|}{|B_k|}$$. Problem is that I fail to construct sequence that would satisfy even the finiteness property, not mentioning the convergence...

Is there a natural way to define probability of random sequence being convergent? And if so, what is it?

• Surely the answer, if any answer could be defined, would be $0$. Why should a random sequence even be Cauchy? If your definition allows for a positive probability that $|a_1-a_0|>\epsilon$, and if you assume the same distribution at each slot independently, then the probability that a random sequence is Cauchy is $0$ (assuming it can be defined). Or did I misunderstand your goal? – lulu Sep 23 '18 at 19:53
• @lulu I didn't think of it in probabilities, but it seems good way to think about it (also the density of odd integers actually makes more sense now to me when I ask what is probability that random number is odd...). I guess you could put that as an answer if you like. – Sil Sep 23 '18 at 19:57
• Not precise enough for an answer, I think. Yes, probabilities are a good approach., I'd say. Your even/odd case works that way. It's sensible to imagine that "probability" there has to mean "limit the choice to $N$ or less, and then let $N\to \infty$. Here, though, I can't think of anything that would give a non-trivial result. – lulu Sep 23 '18 at 20:03
• @lulu I've restated the question in terms of probabilities, but I see your point, probably anything else than $0$ cannot be expected... So it just boils down into how to define it properly... – Sil Sep 23 '18 at 20:19