- #1

StonedPanda

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Let F be the set of expressions of the form a = sum from i in Z of a-sub-i*x*i, where each a-sub-i is an element of R and {i < 0 : a-sub-i does not equal 0) is finite. (X is a formal symbol, not a number). An element a belonging to F is positive if the least indexed nonzero coefficient ak (what does this mean!) in the expression for a is positive. The sum oif a in F and b in F is the element c in F defined by c-sub-i=a-sub-i + b-sub-i for i in Z. the product for a in F and b in F is the element c in F defined by c-sub-j = sum from i in Z of a-sub-i * b-sub-j-i for j in Z.

a) prove that the sum and product of the two elements of F is an element of F.

b) WE have defined addition, multiplication, and order on F. Prove that with these operations, F is an ordered field.

c) Interpret each real number a as the element a in F with a-sub-i = 0 for all i, except a-sub-0 = alpha; this interprets R as a subjset of F. PRove that N is a bounded set in F. COnclude that F does not satisfy the Archimedean property.

I'm a freshman in College and this problem is giving me nightmares. any help at all would be appreciated.