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.
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...
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 +...
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...
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...
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...
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'##...
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...
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...
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##
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?
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...
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...
##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##...
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...
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...
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?
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...
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...
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.
Problem:
Let $\left(X, M, \mu\right)$ be a probability space. Suppose $f \in L^\infty\left(\mu\right)$ and $\left| \left| f \right| \right|_\infty > 0$. Prove that
$lim_{n \rightarrow \infty} \frac{\int_{X}^{}\left| f \right|^{n+1} \,d\mu}{\int_{X}^{}\left| f \right|^{n} \,d\mu} = \left| \left|...
Homework Statement
2. Relevant equation
Below is the definition of the limit superior
The Attempt at a Solution
I tried to start by considering two cases, case 1 in which the sequence does not converge and case 2 in which the sequence converges and got stuck with the second case.
I know...
Hello! (Wave)
I want to find the supremum, infimum of the following sets:
$$\{ x \in \mathbb{R}: 0<x^2-1<3\}, \{1+\frac{(-1)^n}{n}: n=1,2, \dots \}$$
For the first set I have thought the following:
$$ 0<x^2-1<3 \Rightarrow 1<x^2<4 \Rightarrow x^2>1 \text{ and } x^2 <4 \Rightarrow (x>1 \text{...
Hey all,
I started to learn this subject, and i understtod how to find the supremum and infimum of a given set or function.
but I have problem with one question which I can not solve, and I don't know how to start.
This is the quesion:
Given to bounded sets X and Y, which their element are REAL...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with yet a further issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with yet a further issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with another issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with another issue/problem with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
I am reading Houshang H. Sohrab's book: "Basic Real Analysis" (Second Edition).
I am focused on Chapter 2: Sequences and Series of Real Numbers ... ...
I need help with Theorem 2.1.45 concerning the Supremum Property (AoC), the Archimedean Property, and the Nested Intervals Theorem ... ...
Homework Statement
I'm in need of some help to be able to determine the supremum and infimum of the following sets:A = \left\{ {mn\over 1+ m+n} \mid m, n \in \mathbb N \right\}B = \left\{ {mn\over 4m^2+m+n^2} \mid m, n \in \mathbb N \right\}C = \left\{ {m\over \vert m\vert +n} \mid m \in...
Claim: Let A be a non-empty subset of R+ = {x ∈ R : x > 0} which is bounded above, and let B = {x2 : x ∈ A}. Then sup(B) = sup(A)2.
a. Prove the claim.
b. Does the claim still hold if we replace R+ with R? Explain briefly.
So I have spent the past hours trying to prove this claim using the...
1. If x and y are arbitrary real numbers with x < y, prove that there is at least one real z satisfying
x<z<y.2. I'll be using this theorem:
T 1.32 Let h be a given positive number and let S be a set of real numbers. (a) If S has a supremum, then for some x in S we have x > sup S - h.The Attempt...
Homework Statement
Give an example of each, or state that the request is impossible:
1) A finite set that contains its infimum, but not its supremum.
2) A bounded subset of ℚ that contains its supremum, but not its infimum.
Homework EquationsThe Attempt at a Solution
I either understand this...
Homework Statement
Show that the set ##\{x \in \mathbf Q; x^2< 2 \}## has no least upper bound in ##\mathbf Q##; using that if ##r## were one then ##r^2=2##. Do this assuming that the real field haven't been constructed.
Homework Equations
N/A
The Attempt at a Solution
Attempt at proof:
##r\in...
Homework Statement
Let ##A## be a nonempty set of real numbers which is bounded below. Let ##-A## be the set of all numbers ##-x##, where ##x \in A##. Prove that
##\inf A = -\sup(-A)##.
Homework Equations
Definition:
Suppose ##S## is an ordered set, ##E\subset S##, and ##E## is bounded above...
Let S and T be subsets of R such that s < t for each s ∈ S and each t ∈ T. Prove carefully that sup S ≤ inf T.
Attempt:
I start by using the definition for supremum and infinum, and let sup(S)= a and inf(T)= b
i know that a> s and b< t for all s and t. How do i continue? , do i prove it...
I'm trying to deal with the supremum concept in a specific situation, but I think I'm getting the concept wrong.
A step of a proof I'm going through states:
P\ [\sup\limits_{x}\ |f(x)\ -\ f'(x)|\ >\ y\ |\ z]\ \ \leq\ \ \sum_{i=1}^M\ P\ [\ |f(x)\ -\ f'(x)|\ >\ y\ |\ z]\ \ \leq\ \ M\times\...
Homework Statement
Let S \subset \mathbb{R} be bounded above. Prove that s \in \mathbb{R} is the supremum of S iff. s is an upper bound of S and for all \epsilon > 0 , there exists x \in S such that |s - x| < \epsilon .
Homework Equations
**Assume I have only the basic proof...
I am using Spivak Calculus. I have a general doubt regarding the definition of least upper bound of sets.
Let A be any set of real numbers and A is not a null set. Let S be the least upper bound of A.
Then by definition "For every x belongs to A, x is lesser than or equal to S"
Let M be an...
The supremum is defined as the "LEAST" upper bound. The word "least" makes me think, there is a "MOST" upper bound, or at least something bigger than a "least" upper bound.
For a set of numbers, is there anything larger than a supremum? Supremum is analogous to a maximum, but I don't...
If let's say I have an expression:
##x\leq y##
Since the supremum is defined as the "least upper bound," does this make sup(x) for this case ##x=y## or is it ##x = \infty##?
Hi Guys,
I am self teaching myself analysis after a long period off. I have done the following proof but was hoping more experienced / adept mathematicians could look over it and see if it made sense.
Homework Statement
Question:
Suppose D is a non empty set and that f: D → ℝ and g: D →ℝ. If...
Homework Statement : [/B]Let a = sup S. Show that there is a sequence x1, x2, ... ∈ S such that xn converges to a.Homework Equations : [/B]I know the definition of a supremum and convergence but how do I utilize these together?The Attempt at a Solution :[/B] Given a = sup S. We know that a =...
Let $f$ be a mapping of a metric space $M$ into itself. For $A\subset M$ let $\And(A)=sup\left\{d(a,b);a,b\in A\right\}$ and for each $x\in M$, let $O(x,n)=\left\{x,Tx,...{T}^{n}x\right\}$ $n=1,2,3...$
$O(x,\infty)=\left\{x,Tx,...\right\}$ Please prove that...
Mod note: Edited by removing [ sup ] tags.
To the OP: Please don't fiddle with font tags, especially the SUP tag, which renders what you write in very small text (superscript).
Hello everyone
I have just started studying mathematics at university this summer and I have decided to supplement my...
Homework Statement
8. Let ##A## be a non-empty subset of ##R## which is bounded above. Deﬁne
##B = \{x ∈ R : x − 1 ∈ A\}##, ##C = \{x ∈ R : (x + 1)/2 ∈ A\}.##
Prove that sup B = 1 + sup A, sup C = 2 sup A − 1.
The attempt at a solution
Note that ##sup A## exists. Let ##x ∈ B##; then ##x − 1...
$Sup(\sum_{k=n+1}^{\infty}\frac{|x_{k}|^{2}}{4^{k}})$ where
$x=(x_{1},x_{2},...)$ is in $l_{2}$ and the supremum is taken over all $x$ such that $||x||$=1.
I think it is equal to $\frac{1}{4^{n+1}}$ Is this correct?