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0.999 =1 is LUB or exactly the same

  1. Sep 29, 2005 #1
    I want to ask that we say 0.999...=1 because the 1 is the least upper bound (supremum) of 0.999... or are they exactly same numbers.

    I ask this because i am confused with finding (the infimum) the largest lower bound of 1. Do you think it should be 0.999... or is 0.999...=1.

    Thanks for your comments...
     
  2. jcsd
  3. Sep 29, 2005 #2

    TD

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    Homework Helper

    If you have a set [itex]S = \left\{ {0.9,0.99,0.999,0.9999,...} \right\}[/itex] then 1 is indeed the supremum. But [itex]0.999... = 1[/itex], at least if you mean a 0 with an infinite number of decimal 9's.

    You may want to read this topic.
     
  4. Sep 29, 2005 #3
    The terms l.u.b. or g.l.b. are for sets, not individual numbers.

    E.g., the l.u.b. of [0,1) is 1. The g.l.b. of [0,1) is 0.

    The l.u.b. and g.l.b. of {1} is 1.
     
  5. Sep 29, 2005 #4
    1/3= 0.333...

    Can be expressed by the serie

    3/10 + 3/100 + 3/1000...

    1/3 * 3 = 1

    so

    3*(3/10 + 3/100 + 3/1000...)

    9/10 + 9/100 + 9/1000...

    That is 0.999...

    so 0.999... = 1
     
  6. Sep 29, 2005 #5

    HallsofIvy

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    I can't believe this is here! I just responded to a new post in the "archives" about this same topic! 0.9999... is exactly equal to 1!! Yes, 1 is also the supremum of the set {0.9, 0.99, 0.999, ...} but that is exactly the same as saying 0.999... = 1!
     
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