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).

Filter by
Sorted by
Tagged with
2
votes
2answers
43 views

Is my proof here correct (basic real analysis/order theory question)?

I'm self-studying real analysis and came across a simple problem I was trying to solve (although I think this is more of an order theory problem): Let $A\subseteq \mathbb{R}$ so that $ε\:>0$ and ...
4
votes
1answer
69 views

inf sup problem need help

Let $\Bbb{M} $ be the set of decreasing smooth functions in $[0,1]$ for which $f(1)=0.$ Find $$\inf_{f \in \Bbb{M}}\sup_{x \in [0;1]} \frac{x*f(x)}{{\int_{0}^1}f(t)dt} $$ Let now $F(x)=x*f(x)$. I ...
9
votes
0answers
63 views

On an asymptotic improvement of AMM problem 11145 (April 2005)

Motivation Motivated by this question, I tried improve the inequality $$\sum_{k=1}^{n}\dfrac{k}{a_{1}+a_{2}+\cdots+a_{k}}\le2\sum_{k=1}^{n}\dfrac{1}{a_{k}}$$ asymptotically. In other words, with ...
1
vote
5answers
64 views

Prove that $\sup \{ \varepsilon x:\, x\in A\}=\varepsilon \sup A$

Since this is an introductory course to real analysis I am looking for the most simple and direct proof. Based mostly on definitions. Let $\varepsilon$ be a positive real number. If $A$ is a non-...
0
votes
1answer
14 views

Boundedness of functions satisfying some conditions.

Let $I \subset \mathbb{R}$ be an unbounded interval. Assume $\forall a \in I\exists M(a)>0,\, M(a)<\infty$. Can I deduce that $\sup\limits_{a\in I}M(a)<\infty$ as well? Is there any ...
0
votes
1answer
68 views

Prove that $Sup(A + B) = Sup(A) + Sup(B)$

Earlier on in the book it showed that to prove $a = b$ it is often best to show that $a \leq b$ and that $b \leq a$. This is the way I want to go about the proof. I am sure there is an easier way but ...
1
vote
2answers
38 views

Axiom of completeness counterexample for $A\subseteq \mathbb{Q}$

In Abbot's "Understanding Analysis," the Axiom of Completeness is stated as "every nonempty set of real numbers that is bounded above has a least upper bound." He then gives $S = \{r \in \mathbb{Q}| r^...
1
vote
1answer
37 views

Let $f(x)$ be continuous at $x_0$. Let $S(d) = \sup_{(x_0 - d, x_0 + d)} f$, $s(d) = \inf_{(x_0 - d, x_0 + d)} f$. Prove $\lim_{d\to 0}(S - s) = 0$

Sorry for a bit unprecise title, the length is limited. Here is the full problem statement: Let $f(x)$ be a function continuous at $x_0$. Let: $$ S(\delta) = \sup_{(x_0 - \delta, x_0 + \delta)} f ...
-1
votes
1answer
42 views

How to prove two suprema are equal [closed]

Is the following equation correct? How to prove. $$ \sup_{\|x\|=1}\|Ax\|=\sup \frac{\|Ax\|}{\|x\|} $$ Thanks!
1
vote
1answer
25 views

interchange integral and inf [closed]

Let $\mathbb{S}^{n-1}$ be the unit sphere on $\mathbb{R}^n$, A be a positive define diagonal $n\times n$ matrix, and $\mu$ is a probabilistic measure on $\mathbb{S}^{n-1}$. So is it correct to ...
2
votes
4answers
69 views

simple sup norm for a function on a unit circle

From my understanding , the supremum norm would just be the maximum value So if I have a point z that lies on the unit circle $|z|=1$ Question: what would the $\sup_{|z|=1} |2z-1|$ be .... Is it ...
0
votes
1answer
53 views

If $A \subset R$ and there is a lower bound for $A$, then there is greatest lower bound for $A$

Hey I need to prove this Statement using definitions. If $A \subset R$ and there is a lower bound for $A$, then there is greatest lower bound for $A$ My try: I drew a number line a take the sets ...
3
votes
3answers
59 views

$A=\{\frac ab | a,b \in Z^+ , \frac{a^2}{b^2}<2 \}$

Show that the set $$A=\left\{\frac ab | a,b \in Z^+ , \frac{a^2}{b^2}<2 \right\}$$ has a least upper bound $L$ My try: $$\frac{a^2}{b^2}<2$$ $$\frac{a^2}{b^2}<(\sqrt2)^2$$ $$-\sqrt2<\frac{...
1
vote
2answers
52 views

How to prove $\sup(A) \leq \inf(B)$?

I have the following problem: Let $A, B \subseteq\mathbb{R}, A \neq \emptyset$ and $B \neq\emptyset$ such that $$(\forall a \in A \wedge b \in B): a \leq b$$ Prove that $\sup(A) \leq \inf(B)$. ...
0
votes
2answers
50 views

How to find the supremum and infimum of the following set of numbers

Find the supremum and infimum of the set of numbers $\frac{(m+n)^2}{2^{mn}}$, where $m,n$ are natural numbers. I know that lower bound is $0$. How to find the supremum?
1
vote
1answer
54 views

Is $A=\{2^{-n}+1/m: n.m\in \mathbb{N}\}$ bounded? (Supremum, Infimum)

Determine whether the following subsets of $\mathbb{R}$ are bounded above or below. Moreover find the Supremum and Infimum (if they exist) and decide whether it is a maximum or a minimum. Is $A=\{2^{-...
0
votes
0answers
24 views

Exchange order of supremum and infimum

I was struggling a little bit with the following question. Let $A$ and $B$ be two non-empty sets and $f:A \times B \rightarrow \mathbb{R}$. Then it holds $\sup_{y\in Y}\left(\inf_{x\in X}f(x,y)\right)...
1
vote
1answer
37 views

using the definition of a supremum

Could anyone give me some idea's on where to go with this question? The question ask's to use the definition of a supremum, which i know is:
0
votes
1answer
32 views

Supremum and infimum of the sequence $x_n=\frac{(-1)^n}{n}+\sin\frac{n\pi}{2}$

$\left\{x_n\right\}$ is a sequence where $x_n=\frac{(-1)^n}{n}+\sin\frac{n\pi}{2}$. Find $\sup\left\{x_n\right\}$ and $\inf\left\{x_n\right\}$. It is a question in my textbook. The answer is: $\sup\...
0
votes
1answer
25 views

Find $\sup\left\{x_n\right\}$ and $\inf\left\{x_n\right\}$ where $x_n= (-1)^n+\cos\frac{n\pi}{4}$

Let $\left\{x_n\right\}$ be a sequence such that $x_n= (-1)^n+\cos\frac{n\pi}{4}$. Find $\sup\left\{x_n\right\}$ and $\inf\left\{x_n\right\}$. $(-1)^n=-1$ (when $n$ is odd), $1$(when $n$ is even). $\...
1
vote
0answers
72 views

Is $f(z)=a_0+a_1\frac{1}{z}$ bounded on the right half-plane or on the closed planes $\{z\in\mathbb C:\text{Re}(z)\ge\alpha>0\}\subseteq\mathbb C$?

Let $f:D(f)\subseteq\mathbb C\to\mathbb C$ be defined by $$f(z)=a_0+a_1\frac{1}{z},$$ where $a_0,a_1\in\mathbb R$. Problem: Is $f$ bounded on the closed, right half-plane $\{z\in\mathbb C:\text{Re}(z)...
2
votes
1answer
60 views

Calculating the expecation of the supremum of absolute value of a Brownian motion

I got a Brownian motion $B(t)$ that starts in $0$ and want to calculate the expectated value of the supremum on the interval $[0,1]$ of the absolute value of it, i.e. $E \left (\sup \limits_{t \in [0,...
0
votes
2answers
28 views

How to prove a sequence is diverging to infinity using derivative properties?

Let $f(x)$ is continues and has derivative in >$(0,\infty)$ and satisfy $f'(x)>x$ for every $x \in >(0,\infty)$ and let $(a_n)$ be a sequence such that $a_1 = 1$ and for each $n>1$, $a_{n+1} =...
0
votes
2answers
48 views

Prove that $\limsup_{n\to\infty}(a_n)\le \sup(a_n)$ [duplicate]

Let $a_n$ be a bounded sequence. Prove that $$\limsup_{n\to\infty}(a_n)\le \sup(a_n)$$ So I think that the best way to prove it is assume that $\limsup_{n\to\infty}(a_n)> \sup(a_n)$ and then find ...
0
votes
2answers
21 views

If $||x|| < \sup\{||x_i||\}$ and $||x|| > \inf\{||x_i||\}$, then is $x \in \{x_i\}$?

For any set $X \subset \mathbb{R}^d$, is it true that if $$||x|| < \sup\{||x_i|| : x_i \in X\} \quad\text{ and } \quad ||x|| > \inf\{||x_i|| : x_i \in X\}$$ then $$x \in X?$$
0
votes
1answer
49 views

Show that every ordered set with the well ordering has the least upper bound property

Here is a proof attempt: Let $S_a =\{x\in A:x \leq a\}$ (also known as a section of $A$). We firstly prove that $\forall a \in A$, $S_a$ has a supremum in $A$. Clearly, every $S_a$ is bounded above ...
3
votes
2answers
70 views

True of false: $|f(x)-g(x)|<\epsilon$ $\forall x\in I$ $\Rightarrow$ $|\sup(f(x))-\sup(g(x))|\leq \epsilon$ [duplicate]

True of false: $|f(x)-g(x)|<\epsilon$ $\forall x\in I$ $\Rightarrow$ $|\sup(f(x))-\sup(g(x))|\leq \epsilon$ I actually have an idea for a proof in case it is correct, but just to make sure what I'...
0
votes
0answers
24 views

Shapes that maximize gravity at fixed mass and density

Suppose we take the set of all bodies $S\subset\mathbb{R^3}$ for which the limit $$ \lim_{r\rightarrow\infty}m_J(S\,\cap B(0,r)) $$ exists and is a fixed number. Here, $m_J$ denotes the Jordan content ...
0
votes
1answer
63 views

Question regarding supremum of partially ordered sets

I have encountered a question while I was practicing the topic 'upper and lower bound of partially ordered sets'. Let $\mathbb{Q}$ be the set of rational numbers. Let $$ B = \{ x \in \mathbb{...
5
votes
5answers
51 views

Finding the supremum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$ What is the Supremum of the above set? I consider the function $f(x)= x^{\frac{1}{x}}$, and show that $f(x)$ is maximum when $x=e$. ...
2
votes
1answer
43 views

Showing that $\alpha\geq \beta$

Let $\mathcal{B}(F)$ the algebra of all bounded linear operators on a complex Hilbert space $F$. Let $A,B\in \mathcal{B}(F)$. Consider the following numbers: $$\alpha=\sup_{\substack{a,b\in \mathbb{C}...
0
votes
1answer
19 views

Supremum equals maximum on a subset of natural numbers

I'm wondering if $\sup_{x \in M} f(x) = \max_{x \in M} f(x)$ holds when $f$ is some arbitrary function and $M = \{0,1,\dots,n\}$ for some $n \in \mathbb N$. My idea is that $M$ is closed and bounded ...
0
votes
1answer
17 views

What does notation $\inf_{k,l}$ mean for indices $k,l$?

What does notation $\inf_{k,l}$ mean for indices $k,l$? Does it mean that one picks $\inf k$ and $\inf l$ or that one picks some kind of "inf of both $k$ and $l$" (whatever that means)?
0
votes
1answer
19 views

Double supremum of double sequence

This is from "Measures, Integrals and Martingales" by R.L. Schilling, page 28, exercise 4.6ii. Prove that for any double sequence $\beta_{ij}, i, j \in \mathbb{N}$ of real numbers, we have $$ \sup_{...
0
votes
1answer
45 views

Finding the supremum and infimum of $\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$

$\left\{n^{\frac{1}{n}}\;\middle\vert\;n\in\mathbb{N}\right\}$ What is the supremum and infimum of the above set? The set is $\left\{1, 2^{\frac{1}{2}}, 3^{\frac{1}{3}},....\right\}$ Now, $n^{\frac{...
1
vote
2answers
32 views

monotonic function can only have simple discontinuity

I am self-studying Rudin, Principles of Mathematical Analysis. I am having trouble going through the theorem saying that monotonical functions can only have simple discontinuity, i.e., Suppose $f$ is ...
0
votes
2answers
32 views

Is $\inf A $ and $\sup A $ belong $\bar A$?

Let $A$ be a anonempty and bounded subset of $\mathbb{R}$. Now take $A= (0,1)$ in the discrete topology of $\mathbb{R} $. My question is that : Is $\inf A $ and $\sup A $ belong $\bar A$ ? ...
1
vote
1answer
51 views

Supremum of uniformly converges functions goes to the supremum of the limiting function

Let ${ f_n\left(x\right):[0,1] \to \mathbb{R} }$ uniformly convergent to function $f(x)$ which is bounded on the interval $[0,1]$. Prove the following: ${ \lim_{n\to\infty} sup_{[0,1]} f_n\left(x\...
0
votes
1answer
45 views

If $f:[0,\infty)\to\mathbb R$ is continuous and $\tau=\inf\left\{t>0:f(t)>\varepsilon\right\}$, then $f(\tau\wedge t)\le\varepsilon$ for all $t\ge0$

Let $f:[0,\infty)\to\mathbb R$ be continuous, $\varepsilon>0$ and $$\tau:=\inf\left\{t>0:f(t)>\varepsilon\right\}.$$ Are we able to show that $f(\tau\wedge t)\le\varepsilon$ for all $t\ge0$ (...
0
votes
1answer
16 views

For $a,b \in \mathbb{R}$ fixed supremum and infimum property

For $a,b \in \mathbb{R}$ fixed and $S$ is a bounded set above and below I want to prove the following: If $a \geq 0$ then $\inf aS+b = a \inf S +b$ and $\sup aS+b = a \sup S +b$ I am not having ...
0
votes
1answer
138 views

Is supremum / infimum concept an axiom. ( Equivalence between infimum's definitions ) [closed]

Earlier title was : 'Equivalence between infimum's definitions'. But, that changed as failure occurred in proof as shown below. I am sorry, if naive, but this is my experience with proving the part (...
0
votes
0answers
36 views

Is there a way to find the largest $n\in \Bbb N$ s.t. f is n times differentiable?

Is there a method to find sup$\{n\in \Bbb N: \text{f is n times differentiable} \}$?
0
votes
2answers
26 views

Uniform convergence of a sequence of functions 4

Prove that the sequence $\left((nx)/(1+4n^2x^2)\right)_{n\in\mathbb N}$ is not uniformly convergent on $(-a,a)$, where $a > 0$ My attempt: $\lim_{n\to\infty}(nx)/(1+4n^2x^2) = 0 = f(x)$ Now, ...
0
votes
1answer
79 views

Questions based on $\epsilon$ based definition of Supremum.

Need help in vetting my answers for questions in sec 2.5 in chap. 2 (page 8) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. The question ...
0
votes
1answer
64 views

Proving equivalence between $\epsilon$ based & $lub$ definition of supremum.

Based on $\epsilon$ have a new definition of supremum: Let there be a nonempty set $X$ with supremum $s$, then $X\cap(s - \epsilon, s]\ne \emptyset, \,\, \forall \epsilon\gt 0$. The conventional ...
0
votes
1answer
76 views

Supremum proof based on $\epsilon$.

Need help in vetting my answers for questions in section 2.3, 2,4 (on page # 7,8) in chap. 2 (page 7) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. ...
-1
votes
3answers
52 views

Supremum, infimum of $|A|$.

Let $A$ be a nonempty subset of the real numbers. Define the set $|A|$ to be $|A|:= \{|x| : x \in A\}$. If the set $A$ is bounded, is the set $|A|$ bounded? If not, give an example. If so, by what? ...
0
votes
1answer
82 views

Find for given upper-bound, epsilon.

Need help in vetting my answers for Q. 3,4,5 in section 2.2.2 in chap. 2 (page 7) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. Also, this post ...
1
vote
1answer
23 views

Limit of sequence of infimums of a continuous function over converging sequence of sets equals infimum of the function over the limit of the set.

Let $X$ be any topological space. Let $C_n$ be monotone decreasing sequence of compact sets in $X$ that converges to a non-empty compact set $C$. That is $\bigcap_{n\geq 0}C_n = C$ and $C_{n+1}\...
0
votes
1answer
87 views

Find for given supremum, epsilon.

Need help in vetting my answers for Q. 1,2 in section 2.2.2 in chap. 2 (page 7) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. . Q. 1: For ...