  In this paper, we generalize Dranishnikov's asymptotic inductive dimension to
the setting of coarse proximity spaces. We show that in this more general
context, the asymptotic inductive dimension of a coarse proximity space is
bigger or equal to the inductive dimension of its boundary, and consequently
may be strictly bigger than the covering dimension of the boundary. We also
give a condition, called complete traceability, on the boundary of the coarse
proximity space under which the asymptotic inductive dimension of a coarse
proximity space and the inductive dimension of its boundary coincide. Finally,
we show that spaces whose boundaries are $Z$-sets and spaces admitting
metrizable compactifications have completely traceable boundaries.
