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    Simple Supremum Question

    If S=sup {Sn: n>N}, is it true that kS= sup{kSn: n>N} where k is any scalar? Thanks!
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    Unbounded Subset

    If A, B are nonempty subsets of R and A is a subset of B, how can you prove that: if A is unbounded, B is unbounded?
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    Basic Epsilon Question

    Why is it true that: if a+e<b for all e>0 then a≤b? What is the meaning of epsilon here? Thanks!
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    Relationship between inequalities in proofs

    Hi, Could you clarify the relationship between proofs that use ≤ and those that use <? For example, if it's already proven that "abs(b) ≤ a if and only if -a≤ b≤a" can we say this implies that "abs(b) < a if and only if -a< b<a"? It seems that since the first statement holds for all abs(b)...
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    Vector Spaces, Polynomials Over Fields

    Vector Spaces, Polynomials "Over Fields" What does it mean when a vectors space is "over the field of complex numbers"? Does that mean that scalars used to multiply vectors within that vector space come from the set of complex numbers? If so, what does it mean when a polynomial, p(x) is...
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    Function Equivalence

    Homework Statement Two functions in F(S,F) (or going from the vector space S to the vector space F) are equal if and only if they have the same value at each element of S. True or False? Homework Equations How can you prove: if two functions, x and y, are equal then they have the same...