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Questions tagged [supremum-and-infimum]

For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

The supremum (plural suprema) of a subset $S$ of a partially ordered set $T$ is the least element of $T$ that is greater than or equal to all elements of $S$. It is usually denoted $\sup S$. The term least upper bound (abbreviated as lub or LUB) is also commonly used.

The infimum (plural infima) of a subset $S$ of a partially ordered set $T$ is the greatest element of $T$ that is less than or equal to all elements of $S$. It is usually denoted $\inf S$. The term greatest lower bound (abbreviated as glb or GLB) is also commonly used.

Suprema and infima of sets of real numbers are common special cases that are especially important in analysis. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.

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Find supremum and infimum of $(1-1/n^2)^n$

I have to find the supremum and infimum of $(1-1/n^2)^n$ where $n$ is a natural number. A hint is given that an inequality helps. I thought that the inequality which could help is Bernoulli's inequality: $(1+x)^n\ge 1+nx$. But this is not helping.…
eeqesri
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Def. 1.10 Baby Rudin: why doesn't $\mathbb{Q}$ fulfill the definition of having a least upper bound?

I know similar questions have been asked on this site, but I have gone through all that I could find, but I'm still confused. In Baby Rudin, defintion 1.10 states that "An ordered set $S$ is said to have the least-upper-bound property if the…
user5744148
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Calculate sup and inf of $A=\{n/(1-n^2)\mid n>1\}.$

Calculate $\sup$ and $\inf$ of the set, $$A=\{n/(1-n^2)\mid n>1\}.$$ I know that $\sup A=0$ and the $\inf A$ don't exist. I prove that $n/(1-n^2)<0$, but I can't prove that $0-\epsilon
James A.
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How to prove the size of supremum of non-empty subset of R?

I am confused on this: Suppose A is a non-empty set of $\mathbb{R}$ with finite supremum sup A = L. Then how do I prove that $\forall \epsilon>0, \exists a \in A $ s.t $L-a < \epsilon$. Maybe I can assume that $L-a>\epsilon$ or $L-a=\epsilon$ for a…
Jayden Rice
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Highest and lowest, exercise.

Let $ A $ and $ B $ be non-empty sets of $ \mathbb R $ where $ A $ is upper bound and $ B $ is lower bound. Suppose that for every $ \epsilon> 0 $, there exist $ x \in A $ and $ y\in B $ such that $ 0
asd asd
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Supremum and infimum being equal at an interval

Prove / disprove: If $A,B \subset (0,1)$ not empty, and $\inf B = \sup A =: x$ then $A \cap B = \{x\}$ I first tried to disprove by giving a counter-example such as $$ A = \{ \frac1n | n \in \mathbb{N} \setminus \{1\}\} \\ B = ? \\ \sup A =…
CSch of x
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Answer and Proof for Supremum and Infimum questions

I couldn’t find any answer on this 2 questions even I asked several mathematicians. Anybody here who can solve those? https://i.stack.imgur.com/ehLv9.jpg (Sorry for the link, I couldn’t upload the image from my phone)
Mustafa Yiğit Tekin
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Supremum and infimum of two different problems

So I want to calculate the supremum and infimum of $\left\{\ln n\right\}_{n=1}^\infty$ and $\left\{\left(1+\frac{(-1)^n}{2n}\right)^n\right\}_{n=1}^\infty$ separately. For $\left\{\ln n\right\}_{n=1}^\infty$ I am thinking that I should use the…
PythonDaniel
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Prove $\sup \{x^2|x \in S \}$ exists, and equal to $s^2$, given $S \subset \Bbb{R}, S \neq \emptyset$, with sup $s$, inf $t$ and $s \geq -t$

Question Suppose $S$ is a non-empty set of real numbers, with supremum $s$ and infimum $t$, and also that $s \geq -t$. a) Show that $-s \leq x \leq s, \forall x \in S.$ b) Show that $\sup \{x^2|x \in S \}$ exists, and it is equal to $s^2$. Answer a)…
alortimor
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Supremum and maximum

I'm asked to tell whether the supremum and maximum of the following functions exist and to derive them when $x \in B$: $1- B = [0, 4 \pi)~\text{and}~g(x)=\cos x$ $2- B = [0, 4 \pi)~\text{and}~g(x)=\sin^2x+\cos^2x$ For the first one, I found that…
Josué Djossou
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infimum's basic properties in optimization problem

This problem is in Optimizing over some variables slide of Convex Optimization problem. I have a question about basic assumption in this textbook. $$ \inf_{x,y} f(x,y) = \inf_{x} g(x), where, g(x)=\inf_{y} f(x,y) $$ How can I prove it??
kakdodo
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Determine infimum/supremum of the set $\{\frac{n-m}{n+m} \vert n,m \in \mathbb{N}_0\}$

I am going through my analysis-notes from when I was a student. I did not solve the following exercise back then and wanted to give it a try: Determine minimum, maximum, infimum/supremum of the set $C = \{\frac{n-m}{n+m} \vert n,m \in…
Student
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When does principle of iterated supremum fail?

When does $$\text{sup}_{x\in A} \text{sup}_{y\in B}f(x,y) = \text{sup}_{y\in B}\text{sup}_{x\in A} f(x,y)$$ fail? Further can we also have then that $$\text{sup}_{x \in A, y\in B} f(x,y)$$ is greater than both of the expressions above? (By principle…
Tony
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Understanding proof that supremum of lower bounds is infimum

I am trying to understand the proof of "Supremum of lower bounds is infimum": Let A be bounded below, and define B = {b in R : b is a lower bound for A}. Show that sup B = inf A. In particular, I have shown that $\alpha = \sup B$ exists so far, and…
Gareth Ma
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Inf and sup of an intersection and union of 2 sets

I'm looking for an example when there is no supremum of an intersection of 2 sets, also no supremum of an union (not the same example). Each set needs to have a supremum, but not their intersection. Same goes for the union. I'd really appreciate…
curleecurlees
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