Recent content by icestone111

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    Proof Related to the Binomial Theorem

    Oh! Would it work if we just choose b > (A-1)/a? If it is I feel bad for missing something so simple. Looking too hard perhaps.
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    Proof Related to the Binomial Theorem

    Sorry for the slowness... It looks awfully similar to a Cauchy sequence proof. But I do realize (?) that I am trying to solve for b in terms of the given/fixed values a and A. When I look at the "hint" that was given, I keep seeing that (i=2 to b)Ʃ(b choose i)ai ≥ 0 Does this train of thought...
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    Proof Related to the Binomial Theorem

    Thanks for the reply! So, would the a in my problem be equivalent to (1+a), and then you would choose b = 1? Then you would get (1+a)b ≥ 1+ba, And using the expansion formula, they are equivalent when b = 1, so as a result b ≥ 1?
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    Proof Related to the Binomial Theorem

    Homework Statement Use the above to prove that given a rational number a > 1 and A any other rational number, there exists b ε N such that ab > A. Homework Equations The above refers to the proving, by use of both induction and binomial theorem, that (1+a)n ≥ 1+na. Binomial Theorem: (i=0 to...
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    Existence of a Constant n in Integer Function Proof

    Thanks guys, this was a bunch of help. So there was more than expected, and it makes sense now!
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    Existence of a Constant n in Integer Function Proof

    Thanks for your reply, so I do not have to analyze this question any further?
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    Existence of a Constant n in Integer Function Proof

    Homework Statement Let f: Z → Z be a function such that f(a + b) = f(a) + f(b) for all a,b ε Z. Prove that there exists an integer n such that f(a) = an for all a ε Z. The Attempt at a Solution I'm a little bit confused here if this is just supposed to be really simple, or if there's more...
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    Proof: Applications of the Universal Property of Natural Numbers

    I'm just uncertain if my reasoning for part 1 is correct and how to move forward with part 2. I'm pretty sure you do need to prove these (at least part 1) by mathematical induction, I'm just uncertain how to do these inductive steps. That's fine, thanks for taking a look!
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    Proof: Applications of the Universal Property of Natural Numbers

    Ah, sorry about that. f^σ(n) means fσ(n) and fn was meant to be fn
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    Proof: Applications of the Universal Property of Natural Numbers

    Homework Statement N refers to the set of all natural numbers. Part 2: From the previous problem, we have σn : N → N for all n ε N. Show that for any n ε N, σ(n+1)(N) is a subset of σn(N), where we have used n + 1 for σ(n) as we defined in class. 2. The attempt at a solution For Part 2, I...