Let (A,I) be a henselian pair with A Noetherian. Let A^* be the I-adic completion of A. Assume at least one of the following conditions holds

1. A → A^* is a regular ring map,
2. A is a Noetherian G-ring, or
3. (A,I) is the henselization (More on Algebra, Lemma 15.7.10) of a pair (B,J) where B is a Noetherian G-ring.

Given f_1, …, f_m ∈ A[x_1, …, x_n] and a_1, …, a_n ∈ A^* such that f_j(a_1, …, a_n) = 0 for j = 1, …, m, for every N ≥ 1 there exist b_1, …, b_n ∈ A such that a_i − b_i ∈ I^N and such that f_j(b_1, …, b_n) = 0 for j = 1, …, m. See Lemma Tag 0AH5.

Slogan: Approximation for henselian pairs.