What is the measure of the set of sequences with convergent subsequences in the space of all real sequences?

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 $ntoinfty$. 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.