I just want to apologize in advance for just posting the question. I'm sick as a dog, running a temp and can't even figure out what i'm reading and need this done by tomorrow so i'm sorry for not writing any of my ideas with it.
Let A be the set of all infinite sequences of nonnegative integers y = (y_1,y_2,...) having
the property that only a finite number of terms are nonzero. Define the relation R on A by xRy if there is a positive integer n such that x_n <y_n and for all i>n,x_i =y_i.
(a) Prove that R is a total ordering relation on A.
(b) Prove that for each positive integer n there is a section of A that has the same order type as (Z+)n with the dictionary order.
(c) Prove that A with the given ordering ia a well-ordering.