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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 $\sup$ and $\inf$ of $x\sin(\frac{1}{x})$ on $(0,\infty)$.

Find $\sup$ and $\inf$ of $x\sin(\frac{1}{x})$ on $(0,\infty)$. I found an example. Example: Let $f(x)=x\sin(\frac{1}{x})$. We are interested in its behaviour for $x\geq1$. $f'(x)=\sin(\frac{1}{x})-\frac{1}{x}\cos(\frac{1}{x})$.…
곤뇽이조아
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The proof of existence of supremum for a non-empty set which has an upper bound

I was going through a simple proof written for the existence of supremum. When I tried to write a small example for the argument used in the proof, I got stuck. The proof is presented in Vector Calculus, Linear Algebra, And Differential Forms…
whitepanda
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What's wrong with my proof about the largest element of a set?

Suppose there's a (wrong) statement: "Every nonempty subset A of reals that is bounded above has a largest element". I wrongly proved this statement is correct. Where did I get it wrong? Thanks Suppose $\sup A$ is not in $A$. Then $\sup A - a > 0$…
Transylvania
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Find for given set $A$, $|A|$ set's supremum & bounds.

Need help in vetting my answers for the questions in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. Let $A$ be a nonempty subset of the real numbers. Define the set $|A|$ to be $|A|:= \{|x| :…
jiten
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question on the meaning of supremum

I know that supremum means least upper bound. If I have a sequence of events, $\{A_n\}_{n=1}^\infty$ then $$\limsup_{n\rightarrow \infty} A_n = \lim_{n\rightarrow \infty} \sup_{j\geq n} A_j$$ I'm having trouble understanding this statement: "The…
user130306
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Proving that a supremum is unique by contradiction

I am not sure if my proof is proper, so please comment on it and try to fix it if you can. Let $S$ be a non-empty set. Say that a and b are both supremum of $S$ with $a
Valentin
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Finding infimum of a function

I have to show that infimum of the following function $\inf_{y_i}(\lambda_i \|y_i\|_2 + \nu_i^T y_i) = \begin{cases}0 \text{ if } \|\nu_i\|_2 \leq \lambda_i\\ -\infty \text{ otherwise} \end{cases} $, where $y_i \in \mathbb{R}^{n_i}, \lambda_i \in…
user110320
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Why the set $ \mathcal{S} = \left\{x\in\mathbb{Q} : x>0 , x^2<2 \right\} $ doesn't have the least-upper-bound in $\mathbb{Q}$?

Why this set $$ \mathcal{S} = \left\{x\in\mathbb{Q} :x>0 , x^2<2 \right\} $$ does not have a least upper bound in $\mathbb{Q}$ ? I am not seeking for a proof, but rather want to get a clear sense that this set has no least upper bound. I know that…
Leyla Alkan
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Supremum and infimum of Multiplicative inverse

Given $$ f(x)= \begin{cases} 1/x,& 0< x\le 1\\ 0,& x=0 \end{cases} $$ and a partition $P = [0, x_1, x_2,\dots, x_{n-1},1]$ and $I_k = [x_{i-1}, x_i]$, e.g. $I_1 = [0, x_1]$. Is it possible to compute the supremum and infimum of $M_k =…
darkmoor
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Supremum and infimum of $(x + y)(1/x + 1/y)$

I have $$A = \left\{\left(x + y\right)\left(x^{-1} + y^{-1}\right) \mid x,y>0 \right\}$$ and I'm looking for the supremum and infimum of this set. Rewriting the equation, I get $\dfrac{(x - y)^2}{xy} + 4$ so I conclude $\inf \text{A} = 4$. My guess…
Zelazny
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Supremum and infimum nth root of n

I'm looking at $A = \{ \sqrt[n]{n} - {1 \over \sqrt[n]{n}}: n \in \mathbb{N} \}$. I find $inf \text{A} = 0$ and $sup \text{A} = \sqrt[3]{3} - {1 \over \sqrt[3]{3}}$. While the infimum is relatively easy to "see," the supremum is not. Is there a way…
Zelazny
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Supremum property of functions

Let $f$ and $g$ be functions from $\mathbb{R}$ to $\mathbb{R}$. Prove that: a.) $\sup\{f(x)+g(x):x\in\mathbb{R}\}\leq \sup\{f(x):x\in\mathbb{R}\}+\sup\{g(x):x\in\mathbb{R}\}$ b.) $\sup\{f(x)+g(y):x,y\in\mathbb{R}\} = …
Wolfy
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Supremum and Infimum of set

I just got the set and Ι tried to find the supremum and infimum and prove it. $$(x-2)\sqrt{\frac{x+1}{x-1}} \quad \text{ for } \quad 2< x\leq 54$$ I succeed to get to this set $\frac{x-2}{x-1}\sqrt{x^{2}-1}$ but I'm stuck. What can I do now…
NM2
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infimum and supremum

I don't really understand how to find the infimum and supremum and the limsup and liminf of a sequence. $x_n$=$(-1)^n+\frac{1}{n}$ $x_n$=$\frac{(-1)^n}{n}$
ematth7
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Finding sup(S) and inf(S)

Am I doing this right? S={$(-1)^n$|n$\in$natural numbers} I got that S =\begin{cases} -1, & \text{if $n$ is odd} \\ 1, & \text{if $n$ is even} \end{cases} S={$(-1)^n$n|n$\in$natural numbers} I got that S =\begin{cases} -\infty, & \text{if $n$ is…
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