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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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Only one maximum point of $e^{-\frac{2(a+t)}{1-t^2}} -\frac{2(a+t)}{1-t^2} + \frac 1{(1-t)^2}$ on $(-1,0)$

I want to prove that, given $a>1$, the function $$f(t)= \underbrace{e^{-\frac{2(a+t)}{1-t^2}}}_{f_1(t)} \underbrace{-\frac{2(a+t)}{1-t^2} + \frac 1{(1-t)^2} }_{f_2(t)}, \quad -1
QA Ngô
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I have no clue how to show this supremum and infimum theorem

How should I approach this? I understand visually that it makes sense, but I have no idea how to use math regarding supremum and infimum. In general, I am also struggling with these type of proofs because the worse thing is I have no idea where to…
nabu1227
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Comparing supremum and infimum of two sets

Let X,Y⊆ℝ be two non-empty sets. Prove that if ∀x∈X ∀y∈Y x≤y, then supX and infY exist and supX≤infY. in this question, I said pick y'∈Y and since x≤y' ∀x∈X, then X is bounded above and by completeness axiom there exists supX. And for infY i did it…
confused
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Supremum of (1/x+1)

THIS IS PART OF MY HOMEWORK, all I need is some guidance: I am trying to prove that 1 is the supremum of the function $\ f(x)= \frac{1}{(x+1)}$ for every $\ x\gt$ 0. I was able to show that 1 is an upper bound but I can't find a way to show that…
AviAsks
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Exchange order of supremum and infimum

I was struggling a little bit with the following question. Let $A$ and $B$ be two non-empty sets and $f:A \times B \rightarrow \mathbb{R}$. Then it holds $\sup_{y\in Y}\left(\inf_{x\in X}f(x,y)\right)\leq \inf_{x\in X}\left(\sup_{y\in…
Keen
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Supremum of product of sets $A,B$.

Taken from sec. 1.4.1 of the book by Mary Hart, titled: Guide to Analysis. Let $A, B$ be two non-empty sets of real numbers with supremums $\alpha, \beta$ respectively, and let the sets $A + B$ and $AB$ be defined by : $A + B = {a + b: a\in A, b\in…
jiten
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Show that $\alpha+ \beta$ is the supremum of $A + B$.

Taken from sec. 1.4.1 of the book by Mary Hart, titled: Guide to Analysis. Let $A, B$ be two non-empty sets of real numbers with supremums $\alpha, \beta$ respectively, and let the sets $A + B$ and $AB$ be defined by : $A + B = {a + b: a\in A, b\in…
jiten
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Infimum of $n$-th root of $n$

Let $A = \left \{ \sqrt[n]{n} \mid n \in \mathbb{N}\right \}$. I need to find and prove the infimum of $A$. Because $n \in \mathbb{N},$ we can know for sure that $\sqrt[n]{n} \geq 1$. (Does that need a deeper proof?) Then to prove that $1$ is the…
eran-avrahami
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Does $\sup|A-B| \ge |\sup A -\sup B|$?

I have strong intuition that "yes" is the answer, because the difference between any two points between $A$ and $B$ can be only bigger (or equal) to the difference between two specific points. But I couldn't prove it. Would like to see a prove, or a…
Daniel
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Is my intuition correct?

Let $f : X \rightarrow [0,\infty)$ and $A$ , $B\,$ two subsets of $X$ such that $A \cap B \neq \emptyset$. If I have that $ \inf \limits_{X \setminus A} f \,>\, \inf \limits_B f $, does it imply that $\inf \limits_B f = \inf \limits_{A \cap B} f $…
vggls
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Prove property of $A$ implies $\inf A \le 2$

Set $A$ is a proper subset of reals and has a property that for every $a \in A$ there exists $b \in A$ such that $a \ge 2b - 2$. How to prove that $\inf A \le 2$?
user4201961
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Supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$

How to find supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$? I suspect that $\frac{1}{6}$ is infimum and supremum does not exist, but I dont know how to prove it using only the definition of sup/inf.
Q. Xin Lo
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Supremum (pairwise disjoint functions)

Suppose $(f_n)$ is a sequence of pairwise disjoint continuous functions on $[0,1]$ so that $\underset{t\in[0,1]}{\sup}\left\vert f_n(t) \right\vert=1$ for all $n$, then $\underset{t\in[0,1]}{\sup}\left\vert \Sigma a_n f_n(t)…
LanaDR
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Does $M:=\{x \in \mathbb{Q}: x^2<7\}$ $M\subseteq\mathbb{R}$ have an infimum, supremum, min, max?

$M\subseteq\mathbb{R}$ $M:=\{x \in \mathbb{Q}: x^2<7\}$ Does $M$ have an infimum, supremum, min, max? My answer would be that it doesn't because $\sqrt{7}$ and $-\sqrt{7}$ are $\not\in \mathbb{Q}$ Is that correct?
lolan496
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Proof of the Archimedean property

I need to show minimum, maximum, infimum and supremum, if they exist. $$ C:= \bigcup_{n \in \mathbb{N}} [0,1/n[.$$ The Archimedean property says: let $e$, $x$ be real numbers, if $e>0$ and $x>0$ then there exists $n\in \mathbb{N}$ such that…
doniyor
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