An Elementary Proof for the Recurrence of the Product Graph \mathbbZ^2\times\0,1,\ldots,l-1\

It is well known that the two dimensional integer lattice \mathbbZ^2 is recurrent, while the three dimensional integer lattice is transient. In this paper we show that the product graph \mathbbZ^2\times\0,1,\ldots,l-1\ is recurrent. The proof approach only utilizes the elementary methods in probability theory (without the words of electric networks).