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?