Thread: Greatest lower bound and lower bounds

1. 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

2. Originally Posted by nerdo
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, $\displaystyle x\le y$.
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.

3. Originally Posted by HallsofIvy
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, $\displaystyle x\le y$.
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.

4. 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:

$\displaystyle \{x | x\le 0\}.$