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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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supremum can be move to inside function? when function is strictly increasing?

I want to know if $\sup_{x\in X} f(x) = f(\sup_{x\in X}x)$ where $X$ is a compact space, and $f$ is a strictly increasing function: $\frac{df}{dx} >0$ for all $x$.
user1292919
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Comparison Property for Infimum

Can someone let me know whether the following theorem is written correctly? I based it off Apostal. Let $S$ and $T$ be nonempty subsets of $\Bbb{R}$ such that $s\geq t$ for all $s$ in $S$ and $t$ in $T$. Then if $S$ has an infimum, then $T$ has an…
user482939
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how to prove supremum and infimum

I started to learn this subject, and I have understood what It's mean, but I cant find out how to find and to prove it. for example : let $X$ be the set of all rational number in form $m/n$ so $ 0
Moran Tailu
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Finding supremum of a sequence - i can't understand my instructor

I have the sequence: $A = \{ 1 - \frac{1}{n} | n \in \mathbb{N}\}$ I need to find the surpremum with correct mathmatical notation. I got some help from my math instructor, i thought i understood it, but some of the steps i don't understand. I'll…
nammerkage
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infimum (and supremum) proof

I'm trying to find and prove the infimum of the set: $X = ((-1)^n)* 1/(2n+1)$ I assumed that the infimum is : $-1/3$ and proved it like this: 1.$((-1^n)* 1/(2n+1))> -1/3$ (when n is odd),after some math I got that $n>=1$ I needed to show that…
kal pola
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understanding sup notation

From the book, Probability and Measure, Billingsley, 3rd, I have problems understanding Eq. 6.6, Page 88: $$P(\sup_k N_k \leq x ) \leq P(N_n \leq x)$$ where $N_n=1_{A_n}$ and $A_n$ is an iid sequence of events. More specifically, I have problems…
user3889486
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minimum of this simple set

i need again some help here. i am defining the minimum and max and inf and sup of this set $A:=(]1,2[ \cup ]2,3]) \cup \{2\}$ which is equal to the interval $(1,3]$ i say, max is 3, and sup is also 3. and 1 is inf but what is minimum? there are…
doniyor
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Prove supremum=variable? (Homework) Prove that sup(−∞, a) = a. (a is in R)

I'm trying to prove the supremum of a set. In class my teacher went over an example and he proved that sup(−3, 4) = 4. In his proof he used the fact that the supremum had to be positive because he could pick a positive number in the set. But in the…
jack
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When does Max=Sup ?

I know the following statement is true. $\forall S,$ (S is compact and S$\neq\emptyset \to \max(S)=\sup(S))$ In what else situations can be Max=Sup ?
acubens555
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Can we use a limit argument to prove that an infimum = some value?

I want to show that the infimum of the set containing the terms of the harmonic sequence is 0. Can I simply argue that because the harmonic sequence converges to 0 then the infimum of the set containing terms of the harmonic sequence is 0? Our…
user463935
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How to compare sums of infimum (or supremum) of two sets?

$ \sup\{x_n+y_n\}⩽\sup\{x_n\}+\sup\{y_n\} $ $ \sup\{x_n+y_n\}⩾\sup\{x_n\}+\sup\{y_n\} $ $ \sup\{x_n−y_n\}⩽\sup\{x_n\}−\sup\{y_n\} $ $ \sup\{x_n−y_n\}⩾\sup\{x_n\}−\sup\{y_n\} $ $ \sup\{x_n+y_n\}⩽\sup\{x_n\}+\inf\{y_n\} $ $…
faoxis
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Evaluating the supremum of $M_{n}$ for the Weierstrass M-test

I have $$M_{n}=\sup_{x\in\left[1,2\right]}\left|\frac{x}{\left(x+1\right)^{n}}\right|$$ How do I evaluate this? Am I supposed to fix $n\in\mathbb{N}$ before trying to find the supremum? I tried doing that, but I see that if $n=1$, then the supremum…
user281997
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Find an infimum of the function on a set

I came across a nasty task which includes infimum of a function on a set and I am really confused about it. Main confusion is the two arguments of the function. It's not a problem for me in finding infimums and supremums of sets, but here I am just…
petrstr
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Set theory: Supremum and infimum sets proof

Let $A$ be a partially ordered set and let $B \subset A$. Prove that $\upsilon(B)=\upsilon(\lambda(\upsilon(B)))$. $\upsilon(B)$ is the set of all upperbounds of $B$ when $B$ is a subset of $A$. $\lambda(B)$ is the set of all lower bounds of $B$. My…
Cruso James
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Proof of the equality of two suprema?

S is an ordered set and Y a nonempty bounded subset. If X subset Y such that for every y in Y there exists x in X that satisfies y <=x, then supX = supY. How would I go about proving this? My thoughts: Because for every y in Y there exists an x in X…
Sky
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