  We present a discrete multiscale expansion of the lattice potential
Korteweg-de Vries (lpKdV) equation on functions of infinite order of
slow-varyness. To do so we introduce a formal expansion of the shift operator
on many lattices holding at all orders. The lowest secularity condition from
the expansion of the lpKdV equation gives a nonlinear lattice equation,
depending on shifts of all orders, of the form of the nonlinear Schr\"odinger
(NLS) equation
