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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Infimum and Supremum of a set 2^k

I am trying to find the infimum and supremum of a set 2^k where k is an integer. I have determined that as k gets larger, so does 2^k so it is not bounded above and therefore there is no supremum. ...
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Doubt in a step involving triangle inequality

This is a step in page 323 of the book Convex Optimization by Stephen Boyd. I am trying to understand how we move from 2nd to 3rd step. So far, I understood that we have to use triangle inequality and ...
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How to prove the criteria for lower semicontinuity of a function at a point?

I have the following statement I need to prove. Let $f:D \to \mathbb{R}$. Then the function $f$ is lower semicontinuous at some point $x_0 \in D$ if $$\lim_{x \to x_0 } \inf f(x) \geq f(x_0)$$. I have ...
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Taking supremum from integral

By solving Schrodinger equation $H\psi = E\psi$ using Green function I get $$\psi(x) = \int G(x,x')E\psi(x') dx' $$ by taking abs of $\psi$ $$|\psi(x)| \leq E \int |G(x,x')\psi(x')|dx' \leq E \sup_{x'}...
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How to show the mapping from $x$ to its unique best approximation is continuous?

Let $K$ be a compact subset of a normed vector space $(X,\|\cdot\|)$, it's easy to see that, given any $x\in X$, there exists $y\in K$ such that $\|x-y\|=\inf\limits_{z\in K} \|x-z\|$. Now if in ...
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What is the infimum over $x$ of the lagrangian function?

I am learning about duality in convex optimisation. The Lagrangian is defined as $$L(x, \lambda, \nu) = f_0(x) + \sum_{i=1}^m\lambda_if_i(x) + \sum_{i=1}^p\nu_ih_i(x)$$ where suppose the optimisation ...
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Infimum question. [closed]

This is the question. Could someone please show me the working? **Suppose the function g : R → R is continuous and strictly decreasing on R with g(0) = 4, g(1) = 1 and g(2) = 0. Explain why the set S =...
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Greatest lower bound in R

I have a set $$S= \{ x \in \mathbb Q \mid x^2 > 7, x >0\}$$ I have the below statement to show that S has the greatest lower bound $\sqrt 7 \in \mathbb{R}$ $$ \forall x \in S, \exists \sqrt 7 \...
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1answer
67 views

Greatest lower bound in Q [duplicate]

I have a set $$ \{ r \in \mathbb Q \mid r^2 >2, r>0 \}$$ I was wondering why it does not have the greatest lower bound. Isn't $0 \in \mathbb Q$ a greatest lower bound for this set in rational ...
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2answers
63 views

Proving that if a sequence $(a_n)$ is monotonic decreasing and $\lim a_n = l$, then $l=\inf \{a_n:n\in\mathbf{N}\}$

I am self-learning Real Analysis. In proving results about sequences and series of reals, it might useful to use the below fact, so I want to write a rigorous proof for it. But, I'd like to check if ...
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1answer
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Let $S$ be a bounded and nonempty subset of $\mathbb{R}^{+}$. Prove that $\sup(\frac{1}{S}) = \frac{1}{\inf S}, \inf S > 0$

Define $$ \frac{1}{S} = \{\frac{1}{s} | s \in S\}$$. Please check my attempt Proof. Let $\frac{1}{s} \in \frac{1}{S}$. Thus $\inf S \leq s$, hence $\frac{1}{s} \leq \frac{1}{\inf S}$, meaning $\frac{1}...
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$\int f^{-1}$ when $f$ contains jump discontinuities

I have several questions regarding the following problem. Calculus by Michael Spivak, Chapter 13, Problem 21 (3rd Edition) begins as follows: 13-21. Suppose that $f$ is increasing. Figure 16 suggests ...
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20 views

supremum and summation Inequality

I am trying to prove an Inequality $$ \sup_{k \ge 1} \mid x_k \, \mid ^{q-p} \le \, (\mid \sum_{k \ge 1} \mid x_k \mid ^p) ^{q-p \over p} $$ where $1 \le p \le q < \infty$ How should I proceed
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Find the supreme and infimum of the following set, $\left \{\dfrac{3n}{\sqrt{1+2n^2}}: n\in \Bbb N\right \}$.

Find the supreme and infimum of the set $$A=\left \{\dfrac{3n}{\sqrt{1+2n^2}}: n\in \Bbb N\right \}$$ We claim that, $\inf A=\sqrt{3}$, y $\sup A=\dfrac{3\sqrt{2}}{2}=\dfrac{3}{\sqrt{2}}.$ Let's prove ...
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Proving functional space $B(A)$ as Banach space.

I have a functional space as below. Let A be a non empty space and $$ B(A) := \{ f : A \rightarrow \mathbb {R} \lvert \, \lVert f \rVert_\infty : = \sup_{a \in A} \, \lvert f(a) \rvert < \infty \}...
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Help with this proof about sup and inf [duplicate]

I need help with this proof. If x $\in$ A ; y $\in$ B and x$\leq$y prove that sup(A)$\leq$inf(B) My attempt was using this: 0 $\lt$ $\epsilon$ sup(A) - $\epsilon$ $\lt$ x $\leq$ sup(A) ; inf(b) $\leq$...
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1answer
34 views

How to show first property of norm?

I have a normed space X which is the Space of all real-valued Lipschitz continuous functions on [0,1]. The norm is. $$ {\lVert x\rVert}_{lip} =\lvert x(0)\rvert + sup_{s \ne t} \lvert { {x(s) -x(t)} \...
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48 views

Prove the following statements. (a) $\|fg\|_u \le \|f\|_u\|g\|_u$; (b) $\sup_{n\in N} \|f_n\|_u < \infty$

Suppose that $f_n, g_n, f, g:\mathbb R^d→\mathbb R$ are all bounded functions, and suppose further that $f_n→f$ uniformly and $g_n→g$ uniformly. Prove the following statements. (a) $\|fg\|_u \le \|f\|...
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1answer
62 views

Find infimum and closure of $A$

Let $$ \mathbf{A}=\left\{\frac{\mathbf{3 m}+\mathbf{2 n}}{5\mathbf{m}+7 \mathbf{n}}: \mathbf{m}, \mathbf{n} \geq \mathbf{3}\right\} $$ Now find : $inf(A)$ $int(A)$ , interior points $cl(A)$ , ...
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3answers
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Calculate sup and inf of $A=\{n/(1-n^2)\mid n>1\}.$

Calculate $\sup$ and $\inf$ of the set, $$A=\{n/(1-n^2)\mid n>1\}.$$ I know that $\sup A=0$ and the $\inf A$ don't exist. I prove that $n/(1-n^2)<0$, but I can't prove that $0-\epsilon<n/(1-n^...
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1answer
20 views

Sum of a normally distributed rv and a Bernouilli distributed rv

I a trying to understand the following result from this lecture notes. Defining the value at risk as a function of the random variable $L$ and the parameter $\alpha \in (0,1)$: $$ \operatorname{VaR}_{\...
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25 views

Prove linear function is unbounded above on a polyhedron if it cannot reach its maximum.

It seems this is a easy question but I cannot write this proof clearly and concisely. Problem: If $\phi(x)=\sum_{i=1}^nc_ix_i$ cannot reach its maximum on a (closed) polyhedron $P$ (defined by $Ax\leq ...
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Prove that $\sup(AB) = \sup(A)\cdot \sup(B)$ [duplicate]

Let $A, B \subset (0, \infty)$ be non-empty and bounded from above. Construct the set of products: $P(AB) = \{x \in (0, \infty)| \exists a \in A, \exists b \in B \text{ such that }x = ab\}$. Show ...
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1answer
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$\sup\{\frac{1}{f(x)}|x\in[a,b]\} = \frac{1}{\inf\{f(x)|x\in[a,b]\}}$: but is it? [duplicate]

I've been taking a look at some proposed proofs for the following theorem: If $f$ is Riemann integrable in $[a,b]$ and is bounded from zero ($\inf\{|f(x)|,x\in[a,b]\} > 0$), then $\frac{1}{f}$ is ...
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109 views

Prove: If $f$ is integrable, than $|f|$ is integrable

I want to prove this statment: Let $f:[a,b] \to \mathbb{R}$ be a bounded function. Prove that if $f$ is integrable in $[a,b]$ then $|f|$ is also integrable in $[a,b]$ - HINT: first prove that if $M_f=\...
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1answer
28 views

Exchanging the expectation and supremum

Suppose $\epsilon$ is a Rademacher variable such that $\mathrm{P} (\epsilon = + 1) = \mathrm{P} (\epsilon = - 1) = \frac{1}{2}$. And suppose $u \in V \subseteq \mathbb{R}$, I see from some articles ...
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2answers
58 views

Prove that $M_{|f|}-m_{|f|} \le M_f-m_f$

I want to prove this statment: Let $f:[a,b] \to \mathbb{R}$ be a bounded function. Prove that if $f$ is integrable in $[a,b]$ then $|f|$ is also integrable in $[a,b]$ - HINT: first prove that if $M_f=\...
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48 views

Let $x_n\leq y_n$ for each $n\in \Bbb N$. Show that $\liminf x_n \leq \liminf y_n$

Let $x_n\leq y_n$ for each $n\in \Bbb N$. Then, $$\liminf x_n \leq \liminf y_n$$ Here is a solution for the above problem ( source:https://math.stackexchange.com/a/934738/199156) Define $u_n:=\inf\{...
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1answer
37 views

Assistance: “Two functions, prove uniform conv. using sup-norm”

I hope everything is going well. I am interested in demonstrating that for all $n \in \mathbb{N}$ and $x \in B \subseteq [0,\infty)$ the following $$f_n(x) = \frac{x}{1+x^n} \text{ and } g_n(x) = \...
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28 views

Is C_b((a,b)) a complete metric space? [duplicate]

Let $C_b((a,b))$ be the space of continuous and bounded functions on $(a,b) \subset \mathbb{R}$, let $||f||_\infty := \sup\{|f(x)| : x \in (a, b)\}$. Is ($C_b((a,b)),||\cdot||_\infty)$ a complete ...
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59 views

Letting $\varepsilon \to 0$ in proofs

I've seen proofs regarding supremum and infimum of bounded sets in $\mathbb{R}$ involving an arbitrary $\varepsilon > 0$ and 'hard' inequalities (not necessarily strict, but hard to manipulate ...
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2answers
90 views

$\sup(A \cdot B) = \sup A \sup B$

Suppose $A$ and $B$ are bounded and nonempty subsets in $\Bbb{R}$. Define the set $A \cdot B$ as follows $$A \cdot B = \{ab\mid a \in A, b \in B\}$$ I tried proving the statement using inequalities ...
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2answers
50 views

Find $x$ such that $(ax)^{bx}>c$, where $a,b,c,x>0$

Let $a>0$, $b>0$, and $c>0$. Let $$x_0\triangleq \inf\{x>0:{\rm for~all~} \bar x>x, (a\bar x)^{b\bar x}>c\}.$$ What is a good estimate (least-conservative estimate) for $x_0?$
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S is a set of adherent values of a sequence $x_n$

Let a sequence $(x_n)_{n\geq 1}$ be a bounded sequence of real numbers. Define $\mathbf S$ as the adherent set of the sequence. Prove that $\lim \inf\;(x_n)=\inf(S)$. An adherent set $\mathbf S$ is a ...
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prove $\alpha \inf (f(x)) = \sup (\alpha f(x)) \text{ for } \alpha<0$

I'm trying to prove the following as part of bigger proof in my calculus course. $$\alpha \inf f(x) = \sup (\alpha f(x)) \text{ for } \alpha<0$$ So far I have the following: $\forall x: \inf(f(x)) \...
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58 views

Riemann integral $\underline{\int_a^b} \alpha f = \alpha \underline{\int_a^b} f$

During our introduction to Riemann integrals and Darboux sums we recieved we encountered this theorem: Let $f$ be a bounded function in $[a,b]$, Then if $\alpha >0$ then $\underline{\int_a^b} \...
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2answers
42 views

$f(x)=\sup\{|f_n(x)| \ :\ n=1,2,..\}<\infty$ find an example of the sequence $\{f_n\}$ such that, $\sup\{f(x)\ :\ 0<x<1\ \}=\infty$

Let ${f_n}$ be a sequence of real- valued continuous functions on $[0,1]$ such that for every $x\in [0,1]$, we have $f(x)=\sup\{|f_n(x)| \ :\ n=1,2,..\}<\infty$ find an example of the sequence $\{...
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2answers
68 views

Let $K$ be a nonempty closed, convex subset of $R^d$. Prove that if $x\notin K$, then there exists a unique point $y\in K$ that is closest to $x$.

Let $K$ be a nonempty closed, convex subset of $R^d$. Prove that if $x\notin K$, then there exists a unique point $y\in K$ that is closest to $x$. My attempt: Suppose y and z are 2 different point ...
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1answer
42 views

Supremum of a sequence of real numbers

Let $(b_n)_{n\geq 1}$ be a sequence of real numbers whose terms are defined as follows : $b_1 :=x+\frac{22}{13^2} -\frac{2}{13}y$, $b_2 :=x+\frac{22}{14^2} -\frac{2}{14}y$ and so on, where $x \in \...
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1answer
168 views

Prove: $\overline{\int_{a}^{b}} (g+f)(x) \ dx = \int\limits_a^b {{f(x)}dx}+ \overline{\int_{a}^{b}} g(x) \ dx$

I am trying to prove the following theorem: let $f:\mathbb [a,b]\to\mathbb R$ be a Riemann integrable function, and let $g:\mathbb [a,b]\to\mathbb R$ be a bounded function. Prove that: $$\overline{\...
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1answer
56 views

Proving a certain inequality of infimums of upper Riemann integral

An assignment in Riemann's integral: Given two bounded function f and g in [a,b], if I proved that for every partition p: $$ U(f+g,p) \leq U(f,p)+U(g,p) $$ How can I show that: I don't know how to ...
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1answer
38 views

Show that $\sup\ A < \sup\ B \implies \exists b\in B \ |\ b \ \text{is upper bound for} A$

Show that $\sup\ A < \sup\ B \implies \exists b\in B \ |\ b \ \text{is upper bound for} A$ Let $\gamma = \sup\ B - \sup\ A > 0$ and $\epsilon = \frac{\gamma}{2}> 0$ further set $b=\sup\ B- \...
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1answer
39 views

What does it mean for a family of functions $\{f_n\}$ to be bounded?

$\{f_n\}$ is a family of functions continuous on the interval $(0,1)$. I'm forgetting how to define what it means for $\{f_n\}$ to be bounded and my textbook/google searches aren't providing what I ...
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1answer
42 views

Supremum and Continuity for function

Let $f$ be continuous on $[a, b]$. Define a function $g$ as follows: $g(a)=f(a)$ and, for $x$ in $(a, b]$ $$g(x)=\sup \{f(y): y \text { in }[a, x]\}$$ Prove that $g$ is monotone increasing and ...
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1answer
13 views

Supremum over real interval equals supremum over rational 'interval' for continuos functions?

I've recently read the following reasoning in a paper: the mapping $x\mapsto \sup_{s\in[a,b]} f(sx)$, $\mathbb{R} \to \mathbb{R}$, is Borel-measurable, since the supremum over $s\in[a,b]$ equals the ...
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58 views

Prove that $\sup\left (\frac{1}{A}\right )=\frac{1}{\inf A}.$

Let $\varnothing\neq A\subseteq \mathbb R^+$, such that $\inf A>0$, then $$\sup\left (\dfrac{1}{A}\right )=\dfrac{1}{\inf A}.$$ My Try: Notice that $\inf A\leq a$ for all $a\in A$. Then, $1/a\leq 1/...
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1answer
20 views

Show that $\inf_n\tau_n<t$ if and only if $\exists n:\tau_n<t$

Let $(\tau_n)_{n\in\mathbb N}\subseteq\overline{\mathbb R}$, $$\tau:=\inf_{n\in\mathbb N}\tau_n$$ and $t\in\overline{\mathbb R}$. Are we able to show that $$\tau<t\Leftrightarrow\exists n\in\mathbb ...
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2answers
41 views

Find and rigorously prove the supremum of the set:

S = $\{a \in \Bbb{Z} | a < 2+1/2$} Given that a is in the set of all integers ($\mathbb{Z}$), would the supremum of this set be equal to $(2 + 1/2)$? If so, given an upper bound of $(2 + 1/2)$, how ...
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1answer
31 views

Let $A=\{a_k:k \in \mathbb{N}\}$ be a countable set of strictly positive real numbers such that $\mbox{inf}(A) = 0$. Prove an accumulation point.

Let $A=\{a_k:k \in \mathbb{N}\}$ be a countable set of strictly positive real numbers such that $\mbox{inf}(A) = 0$. Prove that $0$ is an accumulation point of A. Must $0$ be the only accumulation ...
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2answers
74 views

The set $\{\frac{1}{x^2-3}: x\in\mathbb{Q}\}$ is bounded? Explain.

The set $\{\frac{1}{x^2-3}: x\in\mathbb{Q}\}$ is bounded? Explain. I got this question some weeks ago in an Introduction to Real Analysis exam. In the exam review, the professor mentioned it could be ...

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