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).

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0
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2answers
31 views

Find $\sup\left\{ \frac{m}{|m| + n }\right\} $ where $m \in \Bbb{Z}$ and $n \in \Bbb{N}$

Im trying to do $\sup A $ and $\inf A$ if $A= \left\{ \frac{m}{|m| + n } : m \in \Bbb{Z} , n \in \Bbb{N}\right\} $ rigoriously. my try: Let $f(m,n) = \dfrac{m}{|m|+n} $. Notice that $f(1,1) = \...
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1answer
45 views

Prove that the supremum of the set $A = \{x \in \Bbb Q \; | x^2 < 2\}$ is $\sqrt{2}$ in $\Bbb R$ [duplicate]

Let $A = \{x \in \Bbb Q \; | x^2 < 2\}$. Prove that $\sup_{x \in A}\, (x) = \sqrt{2}\;$ in real line $\Bbb R.$ I could prove the case in the rational line where no supremum exists. In $\Bbb R, $ ...
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3answers
54 views

If $f,g$ are scalar functions then $\sup\{f(x)-g(y)\}=\sup\{f(x)\}-\inf\{g(y)\}$.

Statement Let be $A\subseteq\Bbb{R}^n$ and let be $f,g:A\rightarrow\Bbb{R}$ functions. So $\sup\{f(x)-g(y)\}=\sup\{f(x)\}-\inf\{g(y)\}$. Clearly $f(x)\le\sup\{f(x)\}$ and $-g(y)\le-\inf\{f(y)\}$ and ...
2
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1answer
58 views

Rudin's POMA Chapter 1 exercise 5

Hi I am writing to check if the proof that I wrote is valid, I feel like it seems kind of right to me but as I am only a beginner in writing proofs I feel like I might've missed something. Question: ...
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0answers
32 views

equivalence of two integral equations

Consider a real function $f: \mathrm{R} \to \mathrm{R}$, $f$ is piecewise right continuous, and this pair of equations: $$ \sup_{x> a} \frac 1 {x-a} \int_a^x f(s) \, ds = \phantom{-}\infty \tag1\\ ...
1
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0answers
20 views

Find the Infimum and supremum of the following sum.

Let integers $n>m>3$ be fixed and define $$S(x_1,...,x_n)=\sum_{k=1}^n\frac{x_k}{x_k+x_{k+1}+...+x_{k+m}},$$ where if the index $k+m$ exceeds $n$ then we define $x_{k+m}=x_{k+m-n}$. I am asked ...
2
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1answer
23 views

integral equality related to Nussbaum functions

Consider a real function $f: \mathrm{R} \to \mathrm{R}$, $f$ is piecewise right continuous, and this pair of equations: $$ \sup_{x> a} \frac 1 {x-a} \int_a^x f(s) \, ds = \phantom{-}\infty \tag1\\ ...
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2answers
20 views

Analysis- Supremum and infimum

I tried to do this by taking $ X=\{1,3,5,7\}$ and $Y =$ set of all odd natural numbers. In this case the inf$(A)$ is negative infinity. And sup$(A)$ is finite. But is it enough to answer the ...
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0answers
37 views

different definitions of Nussbaum functions

I'm trying to understand the difference between two different definitions of the Nussbaum property for real functions. Have a look at Definition 1 ( taken from https://doi.org/10.1016/0167-6911(83)...
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0answers
32 views

Swapping expectation and infimum

Let $\mathcal X$ be a finite alphabet and $X$ be a random variable over $\mathcal X$ with distribution $p_X$. Let also $Y$ be another random variable with some joint distribution. I will use the ...
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1answer
36 views

minimize over a quadratic function

From Boyd & Vandenberghe's Convex Optimization, example 3.5: Suppose the quadratic function: $$f(x, y) = x'Ax + 2x'By + y'Cy$$ (where $A$ and $C$ are symmetric) is convex in $(x, y)$, We can ...
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1answer
15 views

A doubt on the proof of Martingale Convergence Theorem on Jacod-Protter

Theorem: Let $(X_n)_{n\geq1}$ be a submartingale such that $\sup\limits_{n}\mathbb{E}\{X_n^{+}\}<\infty$. Then, $\lim\limits_{n\rightarrow\infty} X_n = X$ exists a.s. (and is finite a.s.). Moreover,...
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0answers
35 views

Doubt on limit of a sequence of random variables

Let's define a probability space $(\Omega$, $\mathcal{F}$, $\mathbb{P})$ and a sequence of random variables $(X_n)_{n\geq0}$ defined on it. If I assume that the limit of such a sequence of random ...
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2answers
24 views

Infimum of an expression, and supremum of another expression

I know that the infimum of a subset (e.g., $\mathbb Z^+)$ of a partially ordered set (e.g., $\mathbb R$) is the greatest element in the subset that is less than or equal to all elements of the ...
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0answers
38 views

Can I find supremum of $(ax^2+bx+c)/x^2$ for non-zero $a, b, c$? Is there any condition for the function to converge?

Actually I think I am wrong. I am trying to find, $$\sup \limits_x \frac{{a{x^2} + bx + c}}{{{x^2}}}$$ I know that the above function is undefined at $x\to0$. I know I cannot use L'hospital rule ...
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1answer
24 views

Why is the fact that the sequence $(M_n)_{n\geq0}$ is increasing shown in the following way? My alternative

I quote Jacod-Protter Given a probability space $\left(\Omega,\mathcal{F},\mathbb{P}\right)$ and an increasing sequence of $\sigma$-algebras $\left(\mathcal{F}_n\right)_{n\geq0}$, let $M=\left(M_n\...
2
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3answers
49 views

$\inf X = \inf\overline{X}$ and $\sup X = \sup\overline{X}$

Let $X \subseteq R$ a bounded set. Prove $\inf X = \inf\overline{X}$ and $\sup X = \sup\overline{X}$. I don´t know how to prove these two statements. I already proved that $A \subseteq B \implies \...
5
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3answers
59 views

How Can You Prove the Completeness of $\mathbb{N}$?

I am trying to self-study real analysis and I am finding it difficult to prove some statements even when I have intuition for it. One exercise in my book asks: Prove that $\mathbb{N}$ is complete. ...
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2answers
22 views

Let $E$ be a bounded subset of $\textbf{R}$, and let $s = \sup(E)$. Show $s$ is an adherent point of $E$, and is also an adherent point of $E^{c}$.

Let $E$ be a bounded subset of $\textbf{R}$, and let $s = \sup(E)$ be the least upper bound of $E$. Show that $s$ is an adherent point of $E$, and is also an adherent point of $\textbf{R}\cap E^{c}$. ...
2
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1answer
31 views

If $X$ is a non-empty subset of $\mathbf{R}$, show that $X$ is bounded if and only if $\inf(X)$ and $\sup(X)$ are finite.

If $X$ is a non-empty subset of $\mathbf{R}$, show that $X$ is bounded if and only if $\inf(X)$ and $\sup(X)$ are finite. MY ATTEMPT Let us prove the implication $(\Rightarrow)$ first. If $X$ is ...
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3answers
30 views

Show that a given map does not induce a metric

I have to show that $d(x,y)=\inf|x_i-y_i|$ is not a metric in $\mathbb{R^n}$. Using the definition $d(x,y)≥0$ is not respected $\forall x,y$ is that enough?
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1answer
22 views

The set I × I (where × denotes the Cartesian product and I = [0, 1]) in the lexicographic order is a linear continuum.

I've found on wikipedia a proof but I don't really understand ot. If a topological space $S$ (order topology) is linear continuum it satisfies the next: a) $S$ has the least-upper-bound property b)...
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1answer
21 views

Is $\underset{y}{\sup} g(x,y)$ continuos in x?

I got a bit confused about the following: Suppose $g: X \times Y \to \mathbb{R}$ is continuos in $x$ and $y$, where $X$ and $Y$ are some metric spaces. I was assuming that then $h : X \to \overline{...
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1answer
28 views

Can the infimum of a strictly positive functional from $l_\infty$ on the nonegative part of the unit sphere in $l_1$ be equal to zero

**Let $p=(p_1,p_2,...)\in l_\infty$ satisfying $p_i> 0$ for all $i$ and $S_+=\{x=(x_1,x_2,...)\in l_1, x_i\geq 0, \forall i, \sum_ix_i=1\}$. Denote $<p,x>=\sum_ip_ix_i$. Is it possible ...
1
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1answer
30 views

upper and lower limits of a number sequence

I've got a number sequence, $ a_n = \frac{1+(-1)^n 2n}{1+3n} $ and I have to calculate the upper limit and the lower limit. First of all, I've divided the sequence in two subsequences: the even ...
1
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1answer
37 views

A monotone function $f$ on $[0,1]$ satisfying $f\big(\frac14\big)f\big(\frac34\big)\lt 0$

Let $f:[0,1]\rightarrow \mathbb{R}$ be a monotonic function with $f\big(\frac14\big)f\big(\frac34\big)\lt 0$ .Suppose sup$\{x\in [0,1]: f(x)\lt 0\}=\alpha$ Which of the following statements are ...
5
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1answer
76 views

Are there any properties of sup?

Can I say: $$\sup |f(x)-h(x)+g(x)-g(x)| = \sup|f(x)-g(x)|+ \sup|g(x)-h(x)|$$ I can't seem to find any properties of $\sup$ , so i am wondering if there is any properties that we can generalize to ...
1
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1answer
36 views

Application of Montel's Theorem

I am working on the following problem: Let $\mathcal{F}$ denote the set of functions which are analytic on a neighborhood of the closed unit disk in $\mathbb{C}$. Find: $$\sup\{|f(0)|\mid f(1/2)=0=...
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3answers
31 views

Infimum and Supremum of sets

In one of my textbooks - A Probability Path, Resnick - the author defines $\inf_{k \ge n}A_k = \cap_{k=n}^\infty A_k$ and $\sup_{k \ge n}A_k = \cup_{k=n}^\infty A_k$. I'm having a hard time ...
0
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4answers
40 views

Let $f$ be continuous on $\mathbb{R}$, and $\inf_\mathbb{R}f(x)<0$. Prove that $\exists c$ such that $f(c)<0$

Problem: Let $f$ be continuous on $\mathbb{R}$, and $\inf_\mathbb{R}f(x)<0$. Prove that $\exists c$ such that $f(c)<0$. Here's where I am: suppose $R$ is the range of $f$. Then, if $\inf_\...
1
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1answer
35 views

Prove that the infimum of set of powers is $0$

Let $A = \{ a^n : n \in \mathbb{N} \}$ and assume $0<a<1$. Prove that $\inf A = 0$ We know that $a > 0$ implies $a^n > 0$ and so $0$ is a lower bound of $A$. Suppose $a^n \geq l$ for ...
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0answers
18 views

Find the supremum,infimum,minimum, and maximum of a function

$f(x) = e^{−1/x} , 𝑥 \in (700,∞) $ Find infimum, supremum, minimum and maximum values of this function.
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1answer
84 views

$\int^1_0f$ to 1 decimal place

Let $f:[0, 1] \to \mathbb{R}$ be an increasing function with $f(0) = 0$ and, for all $n \in \mathbb{Z}^{+}$, $f=1/n$ on $(\frac{1}{n+1},\frac{1}{n}].$ Find $$\int^{1}_{0} f dx$$ to one decimal place. ...
0
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3answers
26 views

Supremum of continuous function on an unbounded interval

Let f(x) be a continuous function on an unbounded interval $[a,\infty)$. If $\lim_{x\to\infty}f(x)=c<1$ and $f(x)<1$ $\forall x\geq a$, can we conclude that $\sup\{f(x)\}<1$? Why or why not?
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1answer
52 views

Proving $\|L\| = \sup\left\{\frac{\|L(x)\|}{\|x\|}\colon x\ne 0\right\}$

Can I please get feedback on my proof below? It is a problem from Bartles' Analysis book. Thank you! $\def\x{{\mathbf x}} \def\0{{\mathbf 0}} \def\N{{\mathbb N}} \def\R{{\mathbb R}}$ The norm of a ...
1
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0answers
19 views

If $\inf_{k\in\mathbb{Z}}|x-y+k|\leq\frac{1}{2N}$, then $\inf_{k\in\mathbb{Z}}|Nx-Ny+k| = N\inf_{k\in\mathbb{Z}}|x-y+k|$.

Suppose that $x,y\in [0,1)$ and define $$d(x,y) = \inf_{k\in\mathbb{Z}}|x-y+k|.$$ Let $N\in\mathbb{N}$ and assume that $d(x,y)<\frac{1}{2N}$, then I want to prove that $$d(Nx,Ny) = Nd(x,y),$$ ...
1
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2answers
44 views

Recursive sequence does not converge

Define $x_n$ recursively as follows: $x_1=1$, $x_{n+1}=x_n+\frac{1}{x_n}$. We are asked to show this sequence is not convergent. Here's my attempt. Since $x_1=1>0$, and for each $n \in \mathbb{N}, ...
0
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1answer
34 views

Show $\sup_{x > a}\frac{\sin x}{x} < 1$

Let $a > 0$. I'm trying to show that $$\sup_{x > a}\frac{\sin x}{x} < 1$$ Of course, showing it is $\leq$ is not hard: one uses that $\sin x < x$ for $x > 0$. Looking at the graph, ...
2
votes
2answers
41 views

Necessity of Archimedean property in construction of Reals?

I am currently self-studying the construction of reals as equivalence classes of rationals. In it, I have read that the Archimedean property is a necessary assumption we have to make to construct $\...
1
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1answer
35 views

If $0\leq x\leq1/2$, then why $\inf_{k\in\mathbb{Z}}|x+k|=x$?

Suppose that $0\leq x\leq1/2$. Then how do I formally prove the (rather intuitive) identity $$\inf_{k\in\mathbb{Z}}|x+k|=x?$$ It is easy to see that $\inf_{k\in\mathbb{Z}}|x+k|\leq x$, even without ...
1
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1answer
19 views

Calculating Lim Sup of a certain expression

I am trying to calculate the following: $$\lim_{||x||\to\infty} \sup_{0 \leq t \leq 2\pi} \cfrac{||f(t,x)||}{||x||^7}$$ where $x = (x_1,x_2,x_3)$ and $f(t,x)= ((x_2^2+x_3^2+1)\cos(t),(x_1^2+x_3^2)\...
1
vote
1answer
38 views

Let $a_{n} = 1/n$. Thus $\sup(a_{n})_{n=1}^{\infty} = 1$ and $\inf(a_{n})_{n=1}^{\infty}$ = 0.

Let $a_{n} = 1/n$. Thus $\sup(a_{n})_{n=1}^{\infty} = 1$ and $\inf(a_{n})_{n=1}^{\infty}$ = 0. MY ATTEMPT Indeed, one has that $a_{n} \leq 1$. Otherwise, we should have \begin{align*} a_{n} > 1 \...
0
votes
1answer
57 views

Supremum Norm Identity

Let $f,g : [0,1] → \mathbb{R}$ be bounded functions. Prove that $ ∥fg∥ ≤ ∥f ∥∥g ∥$ Where we define the sup norm of a bounded function $f : D → \mathbb{R}$ as $∥f∥ := sup${|f(x)| : x ∈ D}$ $. ...
0
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0answers
32 views

Verification exercise Apostol's Calculus, $\mathbb{Q}$ dense in $\mathbb{R}$.

I'm starting studying Calculus from this book and I was asked to prove that there is a rational between any two real numbers $x,y$ with $x<y$. I'm using the following result from the book: Let $h&...
0
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1answer
19 views

distance function between point and compact set

Let be $\left(X,\Vert \cdot \Vert \right)$ a normed space and $\emptyset \neq M\subseteq X$. Consider the function: $dist(\cdot,M):X\to \mathbb{R}$, where $dist(x,M)=\inf\limits_{y\in M}\Vert x-...
0
votes
1answer
43 views

How to prove that the supremum of this set is 0?

it sounds like a silly question, but I'm having trouble to prove that $$ \sup\left(\left\{\left. \frac1n-1 \right|n \text{ is odd}\right\}\right)=0 $$ I'm stuck in proving that $$ \forall \...
0
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0answers
15 views

Differentiable approximation of supremum of a function

I am trying to solve an optimization problem of the form: min$_y$ f(y), s.t. g(y,t)$\leq0$ ∀ t∈D. As I can calculate the gradients for f and g, I was thinking of using a penalty method, in which ...
1
vote
1answer
20 views

Show that in general $\inf \sup \ne \sup \inf$ for bilinear functions

I am working on the following exercise: Let $K \subseteq \mathbb{R}^n$, $L \subseteq \mathbb{R}^m$ and let $F(K,L) \rightarrow \mathbb{R}$ with $$ F(x, \lambda) := c^Tx + \lambda^Tb - \lambda ...
0
votes
0answers
18 views

$f^*$ error using infimum in PAC learning

We are given a collection of models $\mathcal{F}$ from which particular, but arbitrarily chosen models $f$ are sampled and a function $\text{err}(f')$ returns the error in classification of the model $...
2
votes
3answers
46 views

Finding supremum and infimum of a set of rational numbers

Let $f(n,m) = \dfrac{mn}{1+m+n} $. Put $S = \{ f(n,m) : n, m \in \mathbb{N} \} $. Find $\inf S $ and $\sup S$ Attempt: First, it is clear that $f(n,m) \geq 0$ for all $n,m$. So $0$ is a lower bound....

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