  Given two nonempty and disjoint intersections of closed and convex subsets,
we look for a best approximation pair relative to them, i.e., a pair of points,
one in each intersection, attaining the minimum distance between the disjoint
intersections. We propose an iterative process based on projections onto the
subsets which generate the intersections. The process is inspired by the
Halpern-Lions-Wittmann-Bauschke algorithm and the classical alternating process
of Cheney and Goldstein, and its advantage is that there is no need to project
onto the intersections themselves, a task which can be rather demanding. We
prove that under certain conditions the two interlaced subsequences converge to
a best approximation pair. These conditions hold, in particular, when the space
is Euclidean and the subsets which generate the intersections are compact and
strictly convex. Our result extends the one of Aharoni, Censor and Jiang
["Finding a best approximation pair of points for two polyhedra'',
Computational Optimization and Applications 71 (2018), 509--523] who considered
the case of finite-dimensional polyhedra.
