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...
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?
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...
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...
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!
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...