# 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).

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### Find a set A that satisfies the following

Find a set A with a order relation such that: $$\forall a, b,c \in A, \inf({\sup({a,c}),b}) = \sup({\inf({a,b}),\inf({a,c})})$$ It's easy to find a set A of two or one element that satisfies this, but ...
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### A supremum/integral inequality

Let $I$ be an arbitrary index set and for each $i\in I$ let $f_i:\mathbb{R}^n\to [0,\infty)$ be a measurable function. For each $i\in I$ and $n\in \mathbb{N}$ let $f_i^n:\mathbb{R}^n\to [0,\infty)$ be ...
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### Proving that the function $f(x) := |x|^{\frac{\lambda - n}{p}} (1- \psi(x))$ satisfies two specific properties related with limits and supremums.

Let $1 \leqslant p < \infty$ and $0 < \lambda < n$, where $n \in \mathbb N$ is an arbitrary fixed integer that stands for the dimension of the euclidian space $\mathbb R^n$. In everything ...
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### What is the $\inf$ and $\sup$ of the area of quadrilateral given its sides length?

Now asked on MO here. Given the length of the sides of a quadrilateral $a,b,c,d$ the area of the quadrilateral is less than or equal to $\frac{(a+b+c+d)^2}{16}$ i.e it is an upper bound of the area ...
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### A partially ordered set has all suprema iff it has all infima

Let $(P, \leq)$ be a partially ordered set. We will show that every nonempty set bounded above in $P$ has a supremum iff every nonempty set bounded below in $P$ has an infimum. Obviously, it suffices ...
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### Questions About Four Definitions of The Upper and Lower Limits of A Sequence

Related questions have been posted here and here. Background I have seen the following four definitions of the upper and lower limits of a sequence from textbooks and MSE posts: Definition 1$\quad$ [...
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### Riemann Sum with Supremum [closed]

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
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### Riemann Integration and Supremum

Suppose we have $f(t, \mathbf{s}): \mathbb{R} \times \mathbb{R}^{g}$ where $\mathbf{s} \in [0, 1]^{g}$ with $g \in \mathbb{Z}_{+} \cup \infty$, $t \in \mathbf{t}$ where $\mathbf{t}$ is a set of tagged ...
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### Upper bound for supremum of Lipschitz function

Given a Lipschitz function $L$, I want to show the following: For $\gamma >0$, let $\chi(\gamma)$ be an equidistant partition on $[0,1]$ with grid length $n^{-\gamma}$ where $n \ge 1$. Then \begin{...
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### Prove that $\mu_*\leq\mu^*$ where the definition of an inner and outer measure is induced by a measure

I know that similar question has been asked and answered here, here, and here. But I am looking for a different proof based on different definitions. We have the following definition of an outer and ...
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### Showing that the supremum of a given function is finite.

Consider arbitrary elements $n \in \mathbb N$ and $\lambda \in \mathbb R$ such that $0 < \lambda < n$. Problem. My goal is to prove that the supremum $$\sup_{r > 0} f(r)$$ is finite, where ...
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### Application of weak maximum principle.

Fix any open, bounded set $U \subset \mathbb{R}^n$ and suppose that $u \in C^2(U) \cap C(\bar{U})$ is a solution of $$-\Delta u = f \,\,\,\text{in}\,U$$ $$u=g \,\,\,\,\,\,\text{on}\,\,\,\partial U,$$ ...
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### Are these two questions related？

$f$ and $g$ are bounded functions with common domain$D$ $\sup\limits_{x\in D}\big\{f(x)+g(x)\big\}\leqslant\sup\limits_{x\in D}f(x)+\sup\limits_{x\in D}g(x).$ Let both $S$and $T$ be non-empty subsets ...
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### Some doubtful implication for mathematical analysis.

Let, $f(x),g(x),f_1(x),g_1(x)$ are positive real valued bounded and continuous functions on domain of non-negative reals and also having range between $0$ and $1$. And, also, $f_1(x),g_1(x)$ are ...
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### Essentialy bounded functions with compact support are locally integrable and satisfy aditional condition?

Consider arbitrary elements $1 \leqslant p < \infty$ and $0 < \lambda \leqslant n$. Furthermore, during this post I considered the usual Lebesgue measure over $\mathbb R^n$ and I denote the ...
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### Recover an essential infimum given a family of measures from a process,

Suposse that for some family of probability measures $\mathcal{P}$ the process $ess\inf_{\mathbb{P}\in\mathcal{P}}\mathbb{E}^\mathbb{P}[B|\mathcal{F_t}]$ is a submartingale, for some possitive r.v. $B$...
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### If $A,B⊆X⊆ \mathbb R$, $A\cap B=∅$ $A,B$ open then $\sup A<\inf B$ or $\sup B< \inf A$

Let $X\subseteq \mathbb R$ be a metric space. I'm trying to prove that if a subset $I$ is non connected, it implies that $\exists a,b \in I$, $a<b$, $\exists x, a<x<b$ such that $x\notin I$. ...
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### Why isn't $\sup(f(x)+g(x)) = \sup f(x) + \sup g(x)$?

The claim that if $X$ is a non empty set and $f,g:X \to \mathbb{R}$ bounded functions then $$\sup_{x \in X} (f(x)+g(x)) = \sup_{x \in X} f(x) + \sup_{x \in X} g(x)$$ Is not true in general. But why is ...
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