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Homework Help: Show that for every n, n<=x<n+1

  1. Nov 9, 2011 #1
    1. The problem statement, all variables and given/known data
    Assume that for every positive real number x there is an integer n>x.
    Show that for ever real number x there is an integer n such that n<=x<n+1.


    2. Relevant equations
    none


    3. The attempt at a solution
    I think i am supposed to use the well ordered principle of induction to solve this problem so here is my solution.

    Suppose there are two sets S and K and when combined they form the set of all integers.
    Let K contain everything not in S
    if K contains every integer greater than x and S contains the numbers {-∞,..., n-2, n-1, n} then the smallest integer in K is n+1 and therefore
    x<n+1

    if S contains everything that is not in K, then the integers in S can either be less than or equal to x because both situations satisfy the condition that the integer is not greater than x.

    therefore n<=x.
    by the transitivity property we can say that
    n<n+1 and n<=x<n+1.

    i am not sure if i did this correctly and since i am self teaching myself some things in math that i feel i do not have a good basis in, it would be cool if you guys could check my answer.

    thank you in advance
     
  2. jcsd
  3. Nov 10, 2011 #2
    Are you trying to do induction on natural numbers or real numbers? As far as I know, induction isn't used much to prove something for all real numbers, just for all natural numbers.
     
  4. Nov 10, 2011 #3

    HallsofIvy

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    Science Advisor

    Let S be the set of all positive integers greater than x. By the hypothesis, that set is non-empty. Can you show that S contains a smallest member? If so, how does that give you your proof?
     
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