What is Supremum: Definition and 144 Discussions

In mathematics, the infimum (abbreviated inf; plural infima) of a subset

S

{\displaystyle S}
of a partially ordered set

T

{\displaystyle T}
is the greatest element in

T

{\displaystyle T}
that is less than or equal to all elements of

S
,

{\displaystyle S,}
if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.The supremum (abbreviated sup; plural suprema) of a subset

S

{\displaystyle S}
of a partially ordered set

T

{\displaystyle T}
is the least element in

T

{\displaystyle T}
that is greater than or equal to all elements of

S
,

{\displaystyle S,}
if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).The infimum is in a precise sense dual to the concept of a supremum. Infima and suprema of real numbers are common special cases that are important in analysis, and especially in Lebesgue integration. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.
The concepts of infimum and supremum are similar to minimum and maximum, but are more useful in analysis because they better characterize special sets which may have no minimum or maximum. For instance, the set of positive real numbers

R

+

{\displaystyle \mathbb {R} ^{+}}
(not including 0) does not have a minimum, because any given element of

R

+

{\displaystyle \mathbb {R} ^{+}}
could simply be divided in half resulting in a smaller number that is still in

R

+

{\displaystyle \mathbb {R} ^{+}}
. There is, however, exactly one infimum of the positive real numbers: 0, which is smaller than all the positive real numbers and greater than any other real number which could be used as a lower bound.

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1. I Supremum of a set, relations and order

Hello, found this proof online, I was wondering why they defined r_2=r_1-(r_1^2-2)/(r_1+2)? i understand the numerator, because if i did r_1^2-4 then there might be a chance that this becomes negative. But for the denominator, instead of plus 2, can i do plus 10 as well? or some other number...
2. Can You Solve a Problem Using the Definition of Supremum?

For this problem, My solution: Using definition of Supremum, (a) ##M ≥ s## for all s (b) ## K ≥ s## for all s implying ##K ≥ M## ##M ≥ s## ##M + \epsilon ≥ s + \epsilon## ##K ≥ s + \epsilon## (Defintion of upper bound) ##K ≥ M ≥ s + \epsilon## (b) in definition of Supremum ##M ≥ s +...
3. I Showing equivalence of two definitions of essential supremum

Assume ##f: X\to\mathbb R## to be a measurable function on a measure space ##(X,\mathcal A,\mu)##. The first definition is ##\operatorname*{ess\,sup}\limits_X f=\inf A##, where $$A=\{a\in\mathbb R: \mu\{x\in X:f(x)>a\}=0\}$$ and the second is ##\operatorname*{ess\,sup}\limits_X f=\inf B## where...
4. Find the supremum of ##Y## if it exists. Justify your answer.

Refreshing on old university notes...phew, not sure on this... Ok in my take, ##x>0##, and ##\dfrac{dy}{dx} = -3x^2=0, ⇒x=0## therefore, ##(x,y)=(0,\sqrt2)## is a critical point. Further, ##\dfrac{d^2y}{dx^2}(x=0)=-6x=-6⋅0=0, ⇒f(x)## has an inflection at ##(x,y)=(0,\sqrt2)##. The supremum of...
5. Proving the Infimum and Supremum: A Short Guide for Scientists

Hi, I have problems with the proof for task a I started with the supremum first, but the proof for the infimum would go the same way. I used an epsilon neighborhood for the proof I then argued as follows that for ##b- \epsilon## the following holds ##b- \epsilon < b## and ##b- \epsilon \in...
6. On the definition of radius of convergence; a small supremum technicality

I am reading the following passage in these lecture notes (chapter 10, in the proof of theorem 10.3) on power series (and have seen similar statements in other texts): I'm confused about ##|x_0|<R##. If ##M=\sup (A)##, then for every ##M'<M##, there exists an ##x\in A## such that ##x>M'##...
7. I Show ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##

I would wish to receive verification for my proof that ##sup\{a \in \mathbb{Q}: a^2 \leq 3\} = \sqrt{3}##. • It is easy to verify that ##A = \{a \in \mathbb{Q}: a^2 \leq 3\} \neq \varnothing##. For instance, ##1 \in \mathbb{Q}, 1^2 \leq 3## whence ##1 \in A##. • We claim that ##\sqrt{3}## is an...
8. My proof of the Geometry-Real Analysis theorem

Consider a convex shape ##S## of positive area ##A## inside the unit square. Let ##a≤1## be the supremum of all subsets of the unit square that can be obtained as disjoint union of finitely many scaled and translated copies of ##S##. Partition the square into ##n×n## smaller squares (see...
9. A Limits and Supremum: Is It True?

We have ##a_n## converges in norm to ##a## and a set ##S## such that for all ##n\ge 0## $$\sup_{s\in S} <a_n,s><+\infty .$$ Is it true that ##\sup_{s\in S} <a,s><+\infty##
10. MHB Finding the Infimum and Supremum

Hello, I feel like I am struggling with this more than I should. I can tell intuitively what the infimum and supremum are, but I am pretty sure that I need a more formal proof style answer. How would one actually prove this question?
11. Contradictory Proof of Supremum of E: Is it Circular?

N.B. I have inserted the proof here as reference. See the bolded text. My question is, isn't the reasoning "##x^{2}+5 \varepsilon<2,## thus ##(x+\varepsilon)^{2}<2 .## " circular? If we can already find an ##0<\varepsilon<1## such that ##x^{2}+5 \varepsilon<2,## Can't one also claim that " we...
12. I Supremum proof & relation to Universal quantifier

In the following proof: I didn't understand the following part: Isn't it supposed to be : ## a > s_A - \epsilon >0 ## and ## b > s_B - \epsilon >0 ## Because to prove that ## s ## is a supremum, we need to prove the following: For every ## \epsilon > 0 ## there exists ## m \in M ## such...
13. Find the infimum and/or supremum and see if the set is bounded

##S_3 = \left\{ \ x∈ℝ : x^2+x+1≥0 \right\}## I am not sure if I have done this correctly. The infimum/supremum and maximum/minimum are confusing me a bit. This is how I started: ##x^2+x+1=0## ##x^2+x+ \frac1 4\ =\frac{-3} {4}\ ## ## \left\{ x^2+\frac 1 2\ \right\} ^2 +\frac 3 4\ = 0##...
14. MHB Proof of an Infimum Being Equal to the Negative Form of a Supremum ()

Hey guys, I'm kind of in a rush because I'll have to go to my classes soon here at USF Tampa, but I had one last problem for Intermediate Analysis that needs assistance. Thank you in advance to anyone providing it. Question being asked: "Let $A$ be a nonempty set of real numbers which is...
15. MHB Infimum and Supremum of a Set (Need Help Finding Them)

Hey guys, I have this Intermediate Analysis problem that I need help finding the answer to. This is what the question asks: "Find the supremum and infimum of each of the following sets (considered as subsets of the real numbers). If a supremum or infimum doesn’t exist, then say so. No formal...
16. I Is there anything wrong with how the supremum of a set is written?

I'm just having random thoughts today, and I didn't know where to put this, since this isn't even a homework problem. Anyway, is my way of writing the supremum of a set correct syntax-wise, or no?
17. I Application of Supremum Property .... Garling, Remarks on Theorem 3.1.1

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand some remarks by Garling made after the proof of Theorem 3.1.1...
18. MHB Application of Supremum Property .... Garling, Theorem 3.1.1 ....

I am reading D. J. H. Garling's book: "A Course in Mathematical Analysis: Volume I: Foundations and Elementary Real Analysis ... ... I am focused on Chapter 3: Convergent Sequences I need some help to fully understand the proof of Theorem 3.1.1 ...Garling's statement and proof of...
19. Finding an upper bound that is not the supremum

I just want to see if I did this correctly, the interval (0,1) has 2 as an upper bound but the supremum of S is 1. So M would be equal to 2? Thank you.