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Math Help - Greatest lower bound and lower bounds

  1. #1
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    Greatest lower bound and lower bounds

    R={1/n: n in N}

    Cans some one please explain to me why

    1. Set of all lower bounds of R is {x:x in R and x<= 0}

    2. the greatest lower bound is 0.

    Thanks

    Nerdo
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  2. #2
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    Quote Originally Posted by nerdo View Post
    R={1/n: n in N}

    Cans some one please explain to me why

    1. Set of all lower bounds of R is {x:x in R and x<= 0}

    2. the greatest lower bound is 0.

    Thanks

    Nerdo
    Do you know the definitions for these things?

    "Lower bound". x is a lower bound for set Y if and only if, whenever y is in Y, x\le y.
    Every number in that set is clearly positive so every non-negative number (0 and negatives) is less than any number in that set.

    No positive number can be a lower bound. Let x> 0. Then 1/x is also a positive number and there exist an integer n> 1/x (Archimedian property). Then 1/n< x.

    "Greatest lower bound". x is a greatest lower bound for the set Y if it is a lower bound for Y and every other lower bound for Y is less than x. It literaly is the "largest" of all lower bounds.

    Here, the set of lower bounds is the set of nonnegative numbers: [tex]\{x | x\le 0\}. 0 is in that set and is larger than any other number in the set. 0 is the greatest lower bound.
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  3. #3
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    Quote Originally Posted by HallsofIvy View Post
    Do you know the definitions for these things?

    "Lower bound". x is a lower bound for set Y if and only if, whenever y is in Y, x\le y.
    Every number in that set is clearly positive so every non-negative number (0 and negatives) is less than any number in that set.

    No positive number can be a lower bound. Let x> 0. Then 1/x is also a positive number and there exist an integer n> 1/x (Archimedian property). Then 1/n< x.

    "Greatest lower bound". x is a greatest lower bound for the set Y if it is a lower bound for Y and every other lower bound for Y is less than x. It literaly is the "largest" of all lower bounds.

    Here, the set of lower bounds is the set of nonnegative numbers: [tex]\{x | x\le 0\}. 0 is in that set and is larger than any other number in the set. 0 is the greatest lower bound.

    Thank for you explanation, I really appreciate your help for clearing it up for me.

    PS what does [tex]\{x | x\le 0\} mean.
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  4. #4
    Junior Member nimon's Avatar
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    It's part of the code used to get math symbols in your text, but the writer forgot to finish it off. It should say:

     \{x | x\le 0\}.
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