If possible find the supremum and infimum of the following sets.
a) (0, infinity)
b) {1/n: n is an element of the natural numbers}
c) rational numbers AND [sqrt(2), sqrt(3)]
d) {x is an element of the real numbers: x+2>x^2}
e) {1/n-1/m: n,m are elements of the natural numbers}
My answers so far:
a) supremum: none; infimum: 0
b) supremum: none; infimum: 1
c) supremum: none; infimum: none
d) supremum: none; infimum: 0
e) supremum: none; infimum: 0
I haven't had much practice with these, and I would appreciate any feedback!
Hey lovesmath.
Just a question regarding the infininum: does this refer strictly to the greatest lower bound?
The reason I ask is that if you have an open set and one endpoint, then if that end point is the lower point it means that the infinum won't actually exist you will never actually have a greatest lower bound.
You can think of it with say (0,infinity) where you can't have zero (since it's not included) but if you pick any value (say 0.0001) then you can always show there is a lower value (say 0.0000001) but you repeat this forever and you never actually get a fixed value.
http://www.math.ualberta.ca/~bowman/m117/m117.pdf Page 30 at the top.
The only two claims I see in the top half of p. 30 are:
(1) A finite set always has a maximum element;
(2) [0, 1] has maximum element 1, but [0, 1) has no maximum element.
These claims are about a maximum element, not a supremum (or infimum). They are not the same.
The rest of the top half of p. 30 consists of definitions, not claims.
You are saying strange things. In post #7 I asked you to state the following claim more precisely.
You also said the following earlier.
It seems that you are saying in these two quotes that does not exists, which is incorrect. Therefore I asked you to clarify what your claim is, which you have not done. Instead, you referred me to a definition of infimum, which is indeed standard.
Well this is a little silly IMO if the infimum doesn't actually belong to the set (i.e. it's not an element of the set) in general.
It's not you: I've looked at what they consider this so called infimum to be but the idea of having a greatest lower bound where that element doesn't exist seems a misnomer.
I'd see it as more of an actual limit as opposed to an actual fixed bound and it's a little silly, but that's just me.
This is precisely the difference between the infimum and the minimum: minimum is the infimum that belongs to the set.
The name "greatest lower bound" does not suggest that it should be an element of the set. After all, a lower bound is just a number that bounds a set from below. For example, -5 is a lower bound of (0, ∞). The greatest lower bound is just the greatest of those bounds, i.e., the maximum of the set of lower bounds. It is not obvious that it exists because not every set, even bounded from above, has a maximum. Indeed, in rational numbers the greatest lower bound of the set does not exist. It is the completeness property of real numbers that guarantees that this maximum of lower bounds exists.
As an answer to the original question, the supremum of is 1 and the infimum is -1.