Let $A$ be a complete noetherian local ring and $\mathfrak{m}$ be its maximal ideal.

If we have several polynomials $f_i \in A[X_1, \dots, X_m]$ which have a common zero $x_n$ in $A/\mathfrak{m}^n$ for any $n$, then must they have a common zero in $A$? In other words, if we have a scheme $X$ finite type over $A$, does $X(A/\mathfrak{m}^n) \not= \varnothing \ \forall n$ imply $X(A) \not= \varnothing $?

This seems an easy question, but the main problem is compatibility for $x_n$. If $X$ is smooth we conclude by Hensel lemma.

- If $k=A/\mathfrak{m}$ is finite then it's true because inverse limit of non-empty finite sets is non-empty.
- If $A$ is a DVR then we know from a general Hensel lemma from "RATIONAL POINTS IN HENSELIAN DISCRETE VALUATION RINGS" by MARVIN J. GREENBERG.
If $k=A/\mathfrak{m}$ is algebraically closed and uncountable, it is also true by considering constructible topology and the inverse limit of non-empty compact Hausdorff spaces is non-empty.

What about general case? Here we assume that $A$ is a complete noetherian local ring, but this also covers the excellent Henselian local ring case because of Artin approximation.