NL-NWO 1964 Problem 5

Consider a sequence of non-negative integers g $_1,g_2,g_3,...$ each consisting of three digits (numbers smaller than $100$ are also written with three digits; the number $27$ , for example, is written as $027$ ). Each number consists of the preceding by taking the product of the three digits that make up the preceding. The resulting sequence is of course dependent on the choice of $g_1$ (e.g. $g_1 = 359$ leads to $g_2= 135$ , $g_3= 015$ , $g_4 = 000$ ).Prove that independent of the choice of $g_1$ : (a) $g_{n+1}\le g_n$ (b) $g_{10}= 000$ .