For questions on suprema and infima. Use together with a subject area tag, such as (real-analysis) or (order-theory).

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0answers
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prove that $\sup\frac{1}{A}=\frac{1}{\inf A}$ [on hold]

A is a bounded, nonempty set of real numbers. $\inf A\neq0,\frac{1}{A}=\left(\frac{1}{x}: x\in A\right)$ Prove that: $\frac{1}{A}=\frac{1}{\inf A}$
-4
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1answer
36 views

How to prove that $\sup(A\cup B)=\max\{\sup(A),\sup(B)\}$? [on hold]

Let $A$ and $B$ be two bounded, nonempty set of real numbers. Prove that $$\sup(A\cup B)=\max\{\sup(A),\sup(B)\}.$$
0
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1answer
34 views

Prove that $\sup(A-B) = \sup(A) - \inf(B)$

$A-B = \{a-b: a\in A, b\in B\}$. Prove that $\sup(A-B) = \sup(A) - \inf(B)$ OK, let $x=\sup(A), y=\sup(B)$: $a\in A \implies a\leq x$ $b\in B \implies b\leq y$ $a+b\leq x+y$ is a upper bound Take ...
0
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3answers
25 views

Supremum and infimum of set $A=\{\,x \in\mathbb R : (x - a) (x - b) (x - c) (x - d) < 0\,\}$., where $a < b < c < d$

$\sup\{\,x \in\mathbb R : (x - a) (x - b) (x - c) (x - d) < 0\,\}$, where $a < b < c < d$
0
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0answers
22 views

Find supremum and infimum

I have problem with finding sup and inf of set: $\displaystyle \frac{k^2-n}{k^3+n^2}$ where $k,n \in N$ Is taking first n=1 and later k=1 good move ? Because I dont't have other idea
1
vote
1answer
33 views

Proving two numbers from two sets derived from a bounded sequence are equal

Let $(x_i)$ be a bounded sequence. Let $s_i=\sup\{x_j: j\ge i\}$ and $S=\inf\{s_i\}$. Let $L$ be the set of all accumulation points of $(x_i)$. Prove $S=\sup L$. So I know and have shown that ...
0
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1answer
27 views

If $f$ is continuous on a bounded closed interval, then the supremum of $|f|$ is finite

If $f \colon [a,b] \to \mathbb{R}$ is continuous, then $\sup_{x ∈ [a,b]}\left | f(x)\right |$ is finite. Attempt: Suppose $f\colon [a,b] \to \mathbb{R}$ is continuous, then by the Extreme value ...
0
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2answers
31 views

Finding supremum and infimum of a given set and proving it formally

I'm trying to learn how to find supremum and infimum of a given set as this is essential in my further studying. Here's a problem I want to tackle: $A=\{\frac{n-k}{n+k}:n,k\in\Bbb{N}\}.$ Find ...
0
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0answers
25 views

Why is $\inf \lim_{i \rightarrow \infty} a_{n+i}= \lim_{i \rightarrow \infty} \inf a_{n+i} $?

If $A = \{a_n : n \in \mathbb N\}$ represents a bounded sequence and $a_n,a_{n+1},a_{n+2},\cdots $ represent it's sub sequences, then, why is $\inf \lim_{i \rightarrow \infty} a_{n+i}= \lim_{i ...
1
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2answers
50 views

there are only finitely many n with $v+ϵ<x_n$ and $2)$ there are infinitely many $n$ with $v−ϵ<x_n$

Prove that if $v$ is the limit superior of a bounded sequence $X$, then for any $\epsilon>0,$ $(i)$ there are only finitely many n with $v+ϵ<x_n$ and $2)$ there are infinitely many $n$ with ...
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0answers
35 views

Limit Superior and Limit Inferior of sequence

I am taking an introductory course in Real Analysis and $(i)$ My textbook gives the following definition of a limit superior $U$ which satisfies the following conditions: $(a)$ For every ...
0
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1answer
51 views

I think I was wrong: The supremum of a set whose elements squared less than a positive real number is square root of this number?

I think I was wrong... Can you find it out and teach me how to prove (the title)? If $c>0$ a real number. Define $E=\{x\in\mathbb R\mid x^2<c\}$. $E$ is nonempty because $0\in E$ (positivity ...
3
votes
1answer
70 views

Continuity of $a^x$ when it's defined by the ordinary way

I've searched for the discussion of proving the continuity of exponential function, in most cases the function is defined by power series or inverse of log function where the log is defined by ...
0
votes
1answer
16 views

Supremum over a sequence with two indices

Let $(a_{nm})_{n,m=1}^\infty \in \mathbb{R}$. Show that \begin{equation} \sup_{n}\sup_{m}a_{nm} = \sup_{m}\sup_{n}a_{nm} \end{equation} This is a practice question for a test that I'm not really ...
3
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1answer
32 views

S = {x ∈ Q : x² < 2} Prove sup(S) = - inf(S)

Define $S = {x \in Q : x^2 < 2}$. Let $a = \sup(S)$ and $b = inf(S)$. Prove that $a = -b$. (without finding $a$ or $b$) I know $b ≤ x ≤ a \; ∀ x \in S$ and $∀ \epsilon > 0 \; ∃ y_1, y_2$ s.t. ...
0
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4answers
52 views

Supremum and infinum of a set

Find supremum and infimum of the set: $B={ \frac{x}{1+ \mid x \mid }} \ for \ x\in \mathbb{R}$ For me it is visible that it will be 1 and -1 respectively but how to prove it properly?
2
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2answers
34 views

Supremum and Infimum using Archemedian Property

I'm having trouble understanding where the Archemedian Property can be applied when evaluating the supremum and infimum of a set S. For example: Suppose $S = \{\frac{1}{n} - \frac{1}{m} : n,m \in ...
2
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1answer
24 views

The sup{(n-1)/n : n is a natural number} exists?

True or False? a)The sup{(n-1)/n : n is a natural number} exists? b) If {x_n} is a sequence with x_n < 3 for all n, then it cannot converge to 3? For a) True: By definition, the number s is ...
1
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2answers
14 views

Which conditions imply $\sup_n |\ln x_n| < \infty$?

I want to find conditions which imply that $\sup_n|\ln x_n| < \infty$. Intuitively I think that $\inf_n x_n > 0$ and $\sup_n x_n < \infty$ should be enough, but I don't know how to write it ...
2
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2answers
44 views

Real Analysis - Supremums

I'm currently learning about supremums but I'm having trouble understanding them. I understand that for a number M to be the sup(S) it satisfy two conditions: (1) M is an upper bound of S. (2) If M' ...
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1answer
56 views

find sup(A) and inf(A) of set A

Let $\displaystyle A= \left\{\frac{m^2-n}{m^2+n^2} : n,m \in N, m>n \right\}$ find $\sup(A)$ and $\inf(A)$ of set $A$ My idea is to transform $\displaystyle \frac{m^2-n}{m^2+n^2}= ...
0
votes
2answers
32 views

Comparing the Sup and Inf of Two Functions

If $f$ is a function $f$ : $D$ $\rightarrow$ $R$ one says that $f$ is bounded above (resp. bounded below, bounded) if the image of $D$ under $f$ i.e. $f(D)$ = {$f(x) : x\in D$} is bounded above (resp. ...
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1answer
26 views

Supremum of measurable functions

If $\{f_n\}$ is a sequence of measurable functions on the same measurable set then: i) $\sup_{1\le i\le n} f_i$ is measurable for each $n$. ii) $\sup f_n$ is measurable. I don't quite follow the ...
0
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1answer
14 views

show that $\inf \{x_n+y_n:n>N\}\ge \inf \{x_n:n>N\}+\inf \{y_n:n>N\}$

show that $\inf \{x_n+y_n:n>N\}\ge \inf \{x_n:n>N\}+\inf \{y_n:n>N\}$. I am stuck on proving this. I understand it intuitively. My approach is to first do the right part. By defining the ...
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0answers
41 views

Infimum and supremum problems

I have problem with finding inf and sup of sets: $$ A=\left\{ n\in N:\quad 2(-1)^{n+1}+(-1)^{n(n+1)/2} \left(2+\frac{3}{n}\right) \right\}$$ $$ B=\left\{ n\in N: \quad \frac{n-1}{n+1}\cos{\frac{2\pi ...
0
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2answers
48 views

How to show $-\sup(-A)=\inf(A)$?

Let $\emptyset\neq A\subseteq\mathbb{R}$ a bounded set. Consider $-A=\{-a:a\in A\}$. I want to prove that $-\sup(-A)=\inf(A)$. It is easy to see that $-\inf(A)$ is an upper bound of $-A$, so ...
0
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0answers
26 views

Property of limit inferior for continuous functions (second part)

I have the following question (a variation from Property of limit inferior for continuous functions): Let $(x_n)$ a sequence in $X$ and $x\in X$ such that for all $F\in X'$ (the dual space of the ...
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1answer
47 views

Property of limit inferior for continuous functions

I have the following question: Let $(x_n)$ a sequence in $X$ and $x\in X$ such that for all $F\in X'$ (the dual space of the vector space X) we have that $(F(x_n))$ converge to $F(x)$ (that is: the ...
0
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2answers
16 views

A question regarding supremum of bounded sets.

I'm clear with the definition of supremum and bounded sets. But for some reason, this statement in my lecture notes given by my Prof, doesnt seem to make sense. Let $X = [0, 1) ∪ (2, 3]$. In this ...
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3answers
36 views

Infimum of a set with two variables

I have encountered a problem concerning the infimum of a set: Prove that. $$\mathrm {inf} \left\lbrace\sqrt{a^2+{1\over b^2}}:a,b\in(0,1) \right\rbrace=1$$ What I've been able to do is to prove ...
1
vote
1answer
20 views

Finding the convergence radius of a power series

Let $\sum\limits_{n=0}^{+\infty}a_nx^n$ be a power series. Prove that If $\large \lim\limits_{n\to\infty}a_ns^n=0,s>0$, then the power series above converges absolutely for $|x|<s$. ...
0
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1answer
40 views

Infimum and supremum of the set of all numbers whose square is greater than 2

Suppose, $S= \{ x \in \mathbb R\mid x^2 > 2\}$. Then $\sup S = +\infty$. What is $\inf S$? I'm guessing that $\inf S \in (-\sqrt 2, \sqrt 2)$. Is that true?
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3answers
348 views

Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the ...
0
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2answers
28 views

On the definition of Scott Continuity

I somewhere encountered the concept of "Scott Continuity" as follows. Let $P,Q$ be partially ordered sets; a function $f:P\to Q$ is Scott continuous if it preserves directed suprema, i.e. for all ...
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2answers
62 views

Is my proof that $\inf A=-\sup (-A)$ correct?

From baby Rudin (exercise 5 in chapter 1): Let $A$ be a nonempty set of real numbers which is bounded below. Let $-A$ be the set of all numbers $-x$, where $x\in A$. Prove that $$\inf A=-\sup (-A).$$ ...
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1answer
26 views

How do I show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$

I want to show that $\sup_{0\leq x\leq1}|g(x)| \geq \int_0^1|g(x)|dx$ for real-valued $g$ that is continuous for $0\leq x\leq1$ . Is it enough to say that $|g(x)| \leq \sup_{0\leq x\leq1}|g(x)|$ ...
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1answer
52 views

Supremum of sum of two sequences: $\sup (x_n+y_n) \le \sup x_n + \sup y_n$

Prove that $\sup\{x_n+y_n\}\leq \sup\{x_n\}+\sup\{y_n\}$, if both sups are finite. Furthermore, prove that $\limsup\{x_n+y_n\}\leq \limsup\{x_n\}+\limsup\{y_n\}$ if both limsups are finite.
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2answers
22 views

Find exact upper and lower bounds of set

Heloo, i have a set A={x-x^2>0} x belongs to rationals Need to find exact lower and upper bounds of this set. Tried to draw a graphic but it didnt worked. Any hints or solutions will be ...
0
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1answer
41 views

what is the difference between bounded and convergent?

I know that bounded means to have an upper or lower bound. Let $E \subset \mathbb{R}$ be nonempty. The set $E$ is said to be bounded above if and only if there is an $M \in \mathbb{R}$ ...
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3answers
77 views

The supremum of rationals that are less than a given number is equal to that number

I have the following theorem to prove. Given a real number $a$, define the set $S$ such that $S = \{x \in \mathbb Q: x < a\}$. Show that $a = \sup S$. My attempt at a proof is as follows ...
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0answers
13 views

What is the process of finding supremum/infimum of a continuous set?

For example given a set, for constant $k\in \mathbb{N}$ $$ \{ (1 + x^2)^{-k} \ | \ x\in [-1,0) \cup (0,1] \} $$ The supremum is $1$ ( the limit as $x$ approaches $0$), but how does one derive this ...
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1answer
30 views

Deduce supremum of Set A

Define $$Set A=\{1-\frac 1n\ | n \in\ N\}$$A. Deduce sup Ab. Use the quanitifier definition of supremum to prove your conjecture in part (a).My attempt at the solution: I believe sup A is 1?
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2answers
71 views

Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$

Let $A,B \subset \mathbb{R}$. Show that $\sup(A \cup B) = \max\{\sup A, \sup B\}$ Here's what I did so far: Let $\sup A, \sup B$, and $\sup (A \cup B)$ denote upper bounds of $A,B$, and $A \cup ...
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1answer
89 views

For any $n \in \mathbb{N}$ there exists a minimal element in the set $ \{ x\in \mathbb{N}\mid n< x \}$

Problem from V. Zorich Analysis textbook: For any $n \in \mathbb{N}$ there exists a minimal element in the set $ \{ x\in \mathbb{N}\mid n< x \}$ namely $\min\{x \in \mathbb{N}|n<x\}=n+1$* ...
0
votes
1answer
57 views

If $A$ is a subset of $B$, then $\sup A \leq \sup B$

Let $A$ be a subset of $B$, where $A$ is nonempty. How can I show that $\sup A \leq \sup B$? My attempt I said let element $a$ be in $A$, which means $a$ is in $B$ because $A$ is a subset of $B$. I ...
0
votes
3answers
55 views

Help with prove of inf.

I need to prove that: $$\inf\{\frac{1}{3}+\frac{3n+1}{6n^2} \Big| n\in\mathbb N\}=\frac{1}{3}$$ I get stuck with my proof, I'll write it down. $$n\geq1$$ $$3n\geq3$$ $$3n+1\geq4$$ ...
0
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1answer
50 views

Preperation for a test: Show that inf($\frac{1}{n}$)=0 . Please check if what I have is correct

We are given the following definition: If a sequence $(a_n)$ is bounded from below then it has a greatest lower bound for the sequence called a $\textbf{infimum}$. $m=$ infimum of $(a_n)$ if i) ...
0
votes
1answer
71 views

Clarification needed: $\inf (A+B) = \inf A + \inf B$

Let $A$ and $B$ be nonempty bounded subsets of $R$ and let $A+B$ be the set of all sums $a+b$ where $a\in A$ and $b\in B$. Prove $\inf(A+B)=\inf(A)+\inf(B)$ My attempt: Since $A$ and $B$ are ...
0
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0answers
40 views

Properties of Sup, Inf on sets.

I understand this proof is a bit long-winded, but I am only concerned with it is correct or not. It seems sound to me. Claim: If $A,B \subset \mathbb{R}$ and are non-empty, $a \leq b, \forall a \in ...
2
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2answers
75 views

$x+y\sqrt{2}$ infimum ($x,y\in \mathbb{Z}$)

I've looked for help with this question but I have not found anything, I hope this is not a duplicate. Define the set $A=\{\mid x+y\sqrt{2}\mid \ : x,y\in \mathbb{Z}\ \mbox{and} \mid ...