What is the measure of the set of sequences with convergent subsequences in the space of all real sequences? Let $S$ be the set of all unbounded (infinite) real sequences. What is the measure of: the subset $A=$ {sequences that have a convergent subsequence}?
I have a weak understanding of the concept of ”measure”, and there are many different types of “measure”, but I guess I’m referring to probability measure or Lebesgue measure? I’m not sure, but I’m hoping it doesn’t matter. I think I’m referring to the same type of measure that people mean when they say the rationals is measure zero in $\mathbb{R}$.
And I guess the answer is probably either $0$ or $1$ ? Hmm I think it is $1$. Because if I shoot countably many bullets at the real line (but with the holes (points) forming an unbounded set), it seems to me that it’s much more likely there will be a limit point than not. But maybe I am wrong

Baker defined a "Lebesgue measure" on $\mathbb{R}^\infty$, the space of all real sequences which has the following characteristic property: the measure of infinite dimensional rectangle $a_i<x_i<b_i$ is the product $\Pi_i(b_i-a_i)$ when it converges.
Consider the set of all sequences bounded by a constant $|x_i|<M$. Since any bounded sequence has a convergent subsequence it is contained in the OP set. Its Baker's measure is $\infty$ whenever $M>1/2$ because the partial products are $(2M)^n$ and grow without bound when $n\to\infty$. Since the OP set contains these sets for every $M$ its measure is also $\infty$. This should not be surprising. The condition that a sequence has a convergent subsequence does not impose any restrictions on any finite stretch of the sequence. And those stretches are enough to make the measure as large as one wants.
