1. Limited time only! Sign up for a free 30min personal tutor trial with Chegg Tutors
    Dismiss Notice
Dismiss Notice
Join Physics Forums Today!
The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

Homework Help: Apostol's Analysis Problem 1.22

  1. Jan 2, 2012 #1
    1. The problem statement, all variables and given/known data
    Given x>0, suppose a_0 = [x], the largest integer less than or equal to x. Assuming a_0, a_1, . . . , a_n-1 are defined, define a_n as the largest integer such that a_0.a_1...a_n less than or equal to x. Now define r_n = a_0.a_1...a_n, r_0 = a_0. Prove that the sup of the set of all the r_n's is x.

    2. Relevant equations

    3. The attempt at a solution
    I did proof by contradiction but does not work. I've got stuck. Just give me a useful hint, don need full solution. I've analyzed all the meanings carefully, but seems I need a trick to solve this, or maybe not. I even tried to use the notion of limit: I thought about |r_n - x|. It goes 0. But this fact it is even harder to prove.

    Okay, while I'm writing this thread, I've got an idea. Please have a look:
    Suppose there exists an upper bound p of the set such that p is less than x. Then, since |p-x|>0, for some n, 1/(10^n) < |p-x|. Thus p + 1/10^n < x. Note that a_0.a_1...a_n <= p. Thus a_0.a_1...(a_n + 1) < x; but this is a contradiction.

    I've used a huge assumption that "for some n, 1/(10^n) < |p-x|." But I didn't learn this yet in this book. Is there any other way not using it?
  2. jcsd
  3. Jan 4, 2012 #2
    I don't think you haven't copied the problem correctly.
    The book i'm looking at also gives a fixed integer [itex]k\geq2,[/itex] and [itex]a_n[/itex] is defined to be the largest integer s.t. [itex]\displaystyle \sum_{i=0}^n{\frac{a_i}{k^i}} \leq x.[/itex] You should be able to prove that [itex]\sup\{r_n:n\geq 0\}=x[/itex] by first proving that [itex]0\leq a_i \leq k-1, \forall i\geq 1[/itex] and that [itex]r_n\leq r_{n+1},\ \forall n.[/itex] After these I think you can use the idea you got to finish the proof, but don't give [itex]k[/itex] any particular value.
Share this great discussion with others via Reddit, Google+, Twitter, or Facebook