Partitions and (m and n) sums of products

We consider the sums of m increasing products from a sequence 〈Xt〉t = 1∞, denoted SPm(〈xt〉t = 1∞). We show that whenever N is partitioned into finitely many cells, one cell will contain a SPm(〈xt〉t = 1∞) for some sequence 〈xt〉t = 1∞. We show further that given m ≠ n there is a two cell partition of N so that neither cell contains SPn(〈xt〉t = 1∞) ∪ SPm(〈yt〉t = 1∞) for any sequences 〈xt〉t = 1∞ and 〈yt〉t = 1∞.