  Let $\mathfrak g$ be an affine Lie algebra with index set $I = \{0, 1, 2,
\cdots , n\}$ and ${\mathfrak g}^L$ be its Langlands dual. It is conjectured by
Kashiwara et al.([16]) that for each $k \in I \setminus \{0\}$ the affine Lie
algebra $\mathfrak g$ has a positive geometric crystal whose
ultra-discretization is isomorphic to the limit of certain coherent family of
perfect crystals for ${\mathfrak g}^L$. Motivated by this conjecture we
construct a positive geometric crystal for the affine Lie algebra ${\mathfrak
g}= A^{(1)}_n$ for each Dynkin index $k\in I\setminus\{0\}$ and show that its
ultra-discretization is isomorphic to the limit of a coherent family of perfect
crystals for $A^{(1)}_n$ given by Okado et al.([29]). In the process we develop
and use some lattice-path combinatorics.
