"Given an ϵ>0, there exists an N such that: |Ecc-pVNZ−EExact|<ϵ.(1)" is the exact mathematical definition of convergence; this answer is at best a circular argument. — Kanghun Kim Sep 19 at 3:02

Unless one gets to prove that this convergence actually holds — Kanghun Kim Sep 19 at 3:03

First you accepted the answer and gave it an upvote, then you unaccepted it and downvoted it. Interesting. This answer can't be a "circular" argument because nothing is argued, it's just told. You asked whether or not it converges, not for a proof that it converges. — Nike Dattani Sep 19 at 3:41

Aha. Now I get it. — Kanghun Kim Sep 19 at 9:46

I thought I asked to prove it, but I actually asked if such a proof exists. — Kanghun Kim Sep 19 at 9:47

Are the cc/pc/def2 basis sets even defined for arbitrary N? While it's easy to come up with a procedure that generates successively larger basis sets and guaranteeing a convergence to the CBS limit, it's not a trivial question whether a pre-given series of larger and larger basis sets converge to the CBS limit, let alone that the series of cc/pc/def2 basis sets may only be known for a finite number of terms, and there may be no generally accepted way to define them for arbitrary N. — wzkchem5 2 hours ago

@wzkchem5 The cc basis sets are defined for arbitrary N, but in order to optimize the exponents such that they give the lowest possible CISD energy, you need to be able to do the integrals, which while theoretically possible to do, cannot practically be done when the number of orbitals gets too big. I touched on this a bit in my answer when I said that it's quite impractical to regularly go beyond N=10 for triatomics, or N=4 for most molecules of interest, and you're right that even optimizing the exponents of the basis set is very challenging to do beyond N=6, but it's theoretically possible. — Nike Dattani 1 hour ago

@NikeDattani I see. Then the question reduces to: when we add more and more basis functions for each angular momentum and optimize them to give the lowest atomic CISD energy, does the resulting basis converge to the CBS? While numerically this seems to be true to extremely high accuracy, it is not mathematically obvious whether this is rigorously true. And I think the OP is more interested in mathematical rigor than practical feasibility. — wzkchem5 31 mins ago

@wzkchem5 good point again! Would you say that an infinite number of linearly independent 1D Gaussians would form a "basis" for constructing any 1D function with error less than any $\epsilon$? If the answer is yes, then it's not too hard to generalize to 3D or n-dimensions. I was originally thinking that each time we increase $N$ in cc-pV$N$Z we are introducing a new orthogonal basis function, but Susi's answer 4 hours ago says that they're not orthonormal when linear combinations (LCAO) are used for polyatomics. Luckily they don't have to be orthogonal, just linearly independent. — Nike Dattani 20 mins ago

This answer to "Can any function be decomposed as sum of Gaussians?" on StackOverflow is a good place to start! — Nike Dattani 11 mins ago

@NikeDattani There is the theoretical possibility that, when you sort the Gaussians w.r.t. their exponents, the difference between the exponents of two adjacent Gaussians X and Y converges to a non-zero value even though the total number of Gaussians approach infinity. Then the Gaussian basis functions may not be able to exactly expand a Gaussian orbital whose exponent is between those of X and Y. This is an unlikely scenario, but it is not clear to me how to rigorously prove that this won't happen in practice. — wzkchem5 6 mins ago