# 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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### 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-...
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### 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 ...
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### 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 ...
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### 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 ...
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### 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 $$x \in X?$$
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### 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 ...
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### 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'...
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### 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 ...
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### 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 ...
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### 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)?
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### 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 ...
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### 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 (...
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### 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$\{n\in \Bbb N: \text{f is n times differentiable} \}$?
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### 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, ...
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### 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 ...
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### 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 ...
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### 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. ...
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### 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? ...
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### 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 ...
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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 $C_{n+1}\... 1answer 86 views ### Find for given supremum, epsilon. Need help in vetting my answers for Q. 1,2 in section 2.2.2 in chap. 2 (page 7) in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. . Q. 1: For ... 2answers 154 views ### Doubts about supremum. Need help in vetting my answers for Q. 1 in chap. 2 in CRM series book by MAA: Exploratory Examples for Real Analysis, By Joanne E. Snow, Kirk E. Weller. . Let$S_1 = \frac n{n+1} : n \in \mathbb{...
Let $𝑎 \in \mathbb{R}$ and $𝑎<𝑏$. Make a conjecture about the supremum of (a, b). My conjecture was this: Since $𝑎<𝑏$ then $𝑠𝑢𝑝(𝑎,𝑏)=𝑏$. Proof: Let $c = sup(a,b)$. Let $d$ be the ...
In strict lexicographical ordering :Given the following Find, if possible, three explicit upper bounds for $C = \{(x, y) \in \mathbb{R}^2 | y < 0\}$. and write the set of upper bounds for $C$ in ...