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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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"sup" in an equation

I am currently reading through JC Lagarias' "The $3x+1$ Problem and its Generalizations" and have come across some notation reading : $$\sup_{K \ge 0} T^{(K)}(N)$$ Now I assume that this means "suppose that $K$ is greater than or equal to $0$",…
Ben Hortin
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max and min versus sup and inf

What is the difference between max, min and sup, inf?
piotrek
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How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$

If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?
Lona Payne
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Proof that $\inf A = -\sup(-A)$

Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$ So far this is what I have Let $\alpha=\inf(A)$, which allows us to say that…
user77107
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Sum of the supremum and supremum of a sum

Consider two real-valued functions of $\theta$, $f(\cdot): \Theta \subset\mathbb{R}\rightarrow \mathbb{R}$ and $g(\cdot):\Theta \subset \mathbb{R}\rightarrow \mathbb{R}$. Is there any relation between (1) $\sup_{\theta \in \Theta}…
Star
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supremum of expectation $\le$ expectation of supremum?

Suppose that $X$ is an arbitrary random variable, is the following is true for any function $f$: $$\underset{y\in \mathcal Y} \sup \mathbb E\big[f(X,y)\big] \le \mathbb E\big[\underset{y\in \mathcal Y} \sup f(X,y)\big]?$$ If $f$ is convex in $X$,…
syeh_106
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Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the following picture: Picture File:Mandel zoom 00…
Stephan Kulla
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How to deal with lim sup and lim inf?

I am currently taking first course in real analysis following Ross's Elementary Analysis textbook. When I was introduced to lim sup and lim inf, I found it hard to manage to play around or make meaningful conclusions from them because the terms are…
user453616
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Proving rigorously the supremum of a set

Suppose $\emptyset \neq A \subset \mathbb{R} $. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$ This is my attempt: $A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 \leq x_i < 2$ so clearly $2$ is an upper bound. To…
CAF
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Prove that the sum of the infima is smaller than the infimum of the sum

I'm trying to prove the following inequality: Let $f$ and $g$ be bounded real-valued functions with the same domain. Prove the following: $$\inf(f) + \inf(g) \leqslant \inf(f+g).$$ I thought I had proved it, but I made the erroneous assumption…
user64219
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Is $\{\sin^n{(n)}:n\in\mathbb{N}\}$ dense in $[-1,1]$?

I know that it's true that the set $\{\sin{(n)}:n\in\mathbb{N}\}$ is dense in $[-1,1]$ but is the set $\{\sin^n{(n)}:n\in\mathbb{N}\}$ also? I would assume it is but I'm unsure of how to prove this because the $n$th power can change the sign of the…
Peter Foreman
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To show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous

Show that the supremum of any collection of lower semicontinuous functions is lower semicontinuous. Suppose $\{f_n\}$ is a sequence of lower semicontinuous functions on a topological space $X$. Define $$g_k=\sup_{n\ge k}f_n.$$ I could see that…
vnd
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What does the notation inf{...} mean?

I came across $$\inf\{k : f \in C^k\}$$ What does $\inf\{\cdot\}$ mean? I have been looking, but haven't found anything.
Filip
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Convex function can be written as supremum of some affine functions

Let $\phi: \mathbb{R} \to \mathbb{R}$ be a convex function. Prove that $\phi$ can be written as the supremum of some affine functions $\alpha$, in the sense that $\phi(x) = \sup_\alpha \alpha(x)$ for every $x$, where each $\alpha$ is defined…
user362105
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The Supremum and Infimum of a sequence of measurable functions is measurable

I am reading through Folland's Real Analysis: Modern Techniques and Their Applications, and they have the following proposition and proof: Proposition: If $\{f_{j}\}$ is a sequence of $\bar{\mathbb{R}}$- valued measurable functions on $(X,…
möbius
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