Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n − [√

  1. Find the supremum and infimum of S, where S is the set

    S = {√n − [√n] : n belongs to N} .

    Justify your claims. (Recall that if x belongs to R, then [x] := n where n is the largest integer less than or equal to x. For example, [7.6] = 7 and [8] = 8)



    ----I found my infimum to be 0 and my supremum to be 1, but how do i go about proving them? Help please.
     
  2. jcsd
  3. Dick

    Dick 25,810
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    Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    It's pretty obvious that all of the elements in S are in [0,1), right? And you shouldn't have any trouble showing the infimum is 0. Just find a n where f(n)=sqrt(n)-[sqrt(n)] is 0. Showing the supremum is 1 is a little harder. You want to find a sequence of integers a_n such that f(a_n) approaches 1.
     
  4. Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    I actually don’t think an ε proof will work for this, since n must be a natural number, unless you restrict ε to naturals too.

    I’m not sure on this by any means but this is the approach I would take. First because S is a subset of the reals it must have a LUB. Arguing that 1 is an upper bound is easy. I would try to show that if √n − [√n]<1 you can find an √m − [√m] that’s even closer to 1 which would make 1 the smallest upper bound. These three facts together ill show that 1 is the LUB.
     
    Last edited: Oct 3, 2010
  5. You are correct an epsilon argument would not work here.

    Originally, I was going to use density of R. But since there are countably many irrationals in the set proposed it is obvious that I can't use it.

    Since the set proposed is a subset of all irrational numbers between (0,1).


    Your approach is similar to the epsilon argument and I doubt it would work.

    Even the sequence approach suggested is a little hairy as it requires an epsilon argument to show convergence. And one cannot guarantee there are no "jumps". Eg. sqrt(1023) = 31.98437118..... and then sqrt(1024)=32.
     
    Last edited: Oct 3, 2010
  6. Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    Not true since transcendental numbers fall in [0,1)



    But I got it, man did this take me awhile, but my idea can work. I don't wanna give it away, but here's the general idea.

    For contradiction sake, assume that there is a n in N, and for all other m in N that: 1 –(√n − [√n]) < 1 – (√m − [√m]).

    Simplify this, then pick a clever m in terms of n that will get rid of the radicals that cause problems.
     
  7. Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    What exactly about my statement is not true ?

    I said "Since the set proposed a subset of all irrational numbers between (0,1)."

    Transcendentals are irrationals right ?

    I don't see where I went wrong.

    Hmm...
    What exactly are you getting a contradiction from ? I don't follow your argument.


    Using your argument, I suceeded in showing that

    (√n − [√n]) is not bounded above by any number of the form (√m − [√m]). Maybe I am missing something but that doesn't prove suprema.
     
  8. Dick

    Dick 25,810
    Science Advisor
    Homework Helper

    Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    Uh, pick a_n=n^2-1. That's the worst case in some sense. What is [n^2-1]? What's the limit as n->infinity of the difference? And for Simkate, please don't double post again, ok?
     
  9. Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    Either it is too late at night and my brain it shut off or I just don't understand what you mean.

    Did you not say

    f(n) = sqrt(n) - [sqrt(n)]

    Then

    f(a_n) = sqrt(n^2 -1) - [ sqrt(n^2 -1)]



    Seems to me like this may not even converge.

    For some large n we could find f(a_n) =0
     
    Last edited: Oct 3, 2010
  10. Dick

    Dick 25,810
    Science Advisor
    Homework Helper

    Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    Possibly it is too late. f(a_n) is only going to be equal to zero if sqrt(n^2-1) is an integer. What is [sqrt(n^2-1)]? There's a pretty simple answer.
     
  11. Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    I see it . n-1.

    It is defintely too late for me to be thinking :(.


    Your solution works. Hopefully OP can use it.
    In the analysis books I have seen limits appear after suprema and the like.
    Mine is certainly like that.
     
  12. Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    Edit: Oh lol.. I go to bed now. Missed the []
    --------------------------------
    sqrt(n^2 + 1) = n-1 wooot?
    I think he meant that it is never an integer because for it to be an integer it need to be a quadratic number which n^2 + 1 never is.
     
  13. Re: Analysis Question: Find the supremum and infimum of S,where S is the set S = {√n

    We both need sleep

    btw it was
    [sqrt(n^2 -1) ] = n-1

    Haha...sleeeeeeppppppppppp.:)
     
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