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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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A question about SupA used in Lebesgue integration books

There is a classic Lemma about $sup A$ that says: Let $A$ be any subset of the Real Numbers and let $s$ be an upper bound for $A$. $s = supA$ iff for every $\epsilon>0$ there exists $a \in A$ such that $s- \epsilon
Studentmathever
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What does it mean to say that a function assumes its supremum or infimum?

I have to find the following 3 things: A function on interval $[0, 1]$ that does not assume its supremum. A continuous function on $[1, +\infty)$ that does not assume its infimum. A continuous function of $(0, 1)$ that does not assume its supremum…
clockadd
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Find and justify the Supremum of the following set

Find the supremum of the following set: $$A:=\left\{{(n-1)\over(2n+3)} : n \in\mathbb{N}\right\}$$ So I have as my answer that sup(A) = ½ but need to justify. We were taught to justify in two steps, first show our answer is an upper bound for the…
Fkins
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Supremum of the set $\{ x |\cos x < 1/4 \}$ in $\mathbb{R}$

In an exercise we supposed that a number $b$ was the supremum of the set defined in the title. Question: Is $\cos b < 1/4$ or =$1/4$ ? Can anybody enlighten me please ?
Ulysse Touchais
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Infimum and supremum of set $A$

Find the infimum and supremum of the set $A = \{x+y: x, y \in \mathbb{R} \}$ and $x,y$ are real numbers or prove that they do not exist.
Hy L
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Which of the following options are true for continuous function $f$ and $g$ in $[0,1]$?

Let $f$ and $g$ be two continuous function defined on $[0,1]$ such that $\sup f(x)= \sup g(x)$ for $x$ belongs to $[0,1]$ Then: a) There exists $x$ in $[0,1]$ s.t. $f(x)=g(x)$. b) There exists $x$ in $[0,1]$ s.t. $f(x)= g(x)-2$. c) There exists $x$…
Arnabdas
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how do I find supremum of the set

I have a test on Monday and the professor gave a hint about the problem. f : ??? is random How do I solve this problem? Let $f: [-10, 10] \to \mathbb{R}$ be defined by $f(x) = ???$. Let $\Delta$ be the set of positive numbers such that $$\Delta =…
K.SEN
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