A Cantor expansion of a positive integer n is a sumwhere each aj is an integer with 0 ≤ aj...

A Cantor expansion of a positive integer n is a sum

where each aj is an integer with 0 ≤ aj ≤ j and am ≠ 0.

Show that every positive integer has a unique Cantor expansion. (Hint: For each positive integer n there is a positive integer m such that m!≤ n<(m + 1)!. For am, take the quotient from the division algorithm when n is divided by m!, then iterate.)