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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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Characterization of supremum

Let $A\subset(0,\infty)$ non empty. Show that $u=\sup A$ iff: i) $\forall x \in A \quad x \leq u$ ii) $\forall d>1 \quad \exists a \in A: \frac{u}{d}1$, since $u>0$ we…
Lotte
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Find the supremum of a set in terms of another set

Let A be a non empty bounded subset of the real numbers, let $B = A \cap [0,\infty)$ and $C = A \cap (-\infty,0]$. Assume both $B$ and $C$ are non empty. Let $D = \{a^2 : a \in A\}$ Find $\sup D$ in terms of $\sup B$ and $\inf C$ and justify. So my…
Fkins
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Inequality with supreme of linear bounded functions on normed vectorial spaces

Let $X$ and $Y$ be normed vectorial spaces, $X\neq Y$. Let $f:X\rightarrow\mathbb{R}$ and $g:Y\rightarrow\mathbb{R}$ be linear bounded functions. Is true the following inequality? $$\boxed{\displaystyle\sup_{x\in X}\dfrac{f(x)}{\|x\|}+\sup_{y\in…
yemino
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Supremum property of a family of functions

Let $\{f_n\}_{n\in\mathbb{N}}$ be a family of functions from $\mathbb{R}$ to $\mathbb{R}$. a.) $\sup\{\sum_{n}f_n(x):x\in\mathbb{R}\}\leq \sum_{n}\sup\{f_n(x):x\in\mathbb{R}\}$. b.) $\sup\{\sum_{n}f_n(x_n):x_n\in\mathbb{R}\} =…
Wolfy
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If S = {1/n | n ∈ ℕ}, what is inf(S)?

If $S = \{1/n \mid n ∈ ℕ\}$, what is $\inf(S)$? I believe the answer is $0$, but I'm not really sure how to prove it...does it involve using epsilon?
Jizibel
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Difference between sup and max

I am working at some Fuzzy-Logic and I am having my problems with the inferece. While using the generalised modus ponens you are using this formula μB'(y) := sup{min(μA'(x),min(μA(x),μB(y))) | x∈X} for y∈Y My Question is, where is the Difference…
yoko
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supremum of $A= \bigcap_{n \ge 1} \left(0, 1 + \dfrac{1}{n}\right)$

Can someone help me . I need to find the supremum of $A= \bigcap_{n \ge 1} \left(0, 1 + \dfrac{1}{n}\right)$. I know it is an interval but then i dont know .
jan
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Confused about my professor's answer for an inf and sup question

For $x_n=\frac{(-1)^n}{n}$, my professor said that the inf for when n is even is $\frac{-1}{n+1}$ and the sup is $\frac{1}{n}$. How is the inf a negative number since $(-1)^n$ where n is an even number is always positive?
ematth7
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supremum and infimum of set $\left\{ \frac{nk}{1+2n+3k} : n,k \in \mathbb{N} \right\}$

First of all, I can't use limits. I need to use $0 < \epsilon < \epsilon_{0}$ method. So when $n=1$, I get $\frac{k}{3+3k} = \frac{1}{3} - \frac{1}{3(1+k)}$ and we all know that for very large k the limit is $\frac{1}{3}$ so because $0 < \epsilon <…
tomtom
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Why may the supremum of a piecewise continuous function not be one of the $f(x)$ values for an open interval of $x$?

The supremum of a function $f(x)$, where $x$ is a member of the closed interval $[a,b]$, is one of the values assumed by $f(x)$ in the $x$-interval $[a,b]$ So, if $f(x)$ is piecewise continuous over an open interval of $x$, is sup $f(x)$ not one of…
Tuhin Paul
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Prove that the greatest area which the $\Delta APY$ can have is $3\sqrt3\frac{a^2}{8}$sq units

From a fixed point $A$ on the circumference of a circle of radius $a$,let the perpendicular $AY$ fall on the tangent at a point $P$ on the circle,prove that the greatest area which the $\Delta APY$ can have is $3\sqrt3\frac{a^2}{8}$sq units. I…
Vinod Kumar Punia
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A property of infimum??

Let $X$ be some space (eg. vector space or Banach space). When is it true that: for any $\epsilon >0$ small, there exists an $f \in X$ such that $$(1+\epsilon) \inf_{g \in X} I(g) \geq I(f)?$$ Here $I:X \to \mathbb{R}$.
EasyStarter
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Let $A= [0,1] - \{1/n │n \in \mathbb{N}\}$. Find $\sup(A)$, $\inf(A)$, $\min(A)$, $\max(A)$.

My idea of this question is to claim $\sup(A)$ and $\inf(A)$ exists (and equals a value) and prove by contradiction that $\min(A)$,$\max(A)$ exists afterwards (and equals $\sup(A)$,$\inf(A)$). The issue that I have is how to interpret set $A$... I'm…
John B.
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Equality of sets supremum and infimum

Suppose we have a right-continuous function $f$ defined on $[0,\infty)$. I now would like to prove if $\lambda >0$ and $t\geq 0$ we have that \begin{equation} \left\{ \inf \{ s\geq 0 : f(s) \geq \lambda \} \leq t \right\} = \left\{ \sup_{s\leq t}…
user155670
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Supremums of sequences

Let $x_n$ and $y_n$ be two sequences of real numbers. Assume that $y_n$ is bounded above and that $x_n$$<$$y_n$ for all n$\in$$\mathbb{N}$. (a) Prove that $x_n$ is also bounded above. (b) Prove that sup($x_n$)$\leq$sup($y_n$). (c) Do the…
Ldog327
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