Absolute Value Operations and the Non-Closure of Profinite Eventually Periodic Sequences

Termwise addition, multiplication, and integral polynomial operations preserve profinite eventually periodic (RFEP) sequences; the termwise absolute value does not. We give an explicit counterexample: let s_n = 1 precisely when n = 2^k with k >= 2 and s_n = -1 otherwise, and put a_n = 1 + s_n n!. Since m divides n! for n at least m, one has a_n congruent to 1 modulo m, so A is RFEP with T(A)(m) = 1; but for n at least 3 the residue |a_n| modulo 3 is exactly 1 or 2, re-encoding the non-eventually-periodic sequence s, so |A| is not RFEP.