# Tagged Questions

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

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### inf e sup of empty set?

If $S$ is an ordered set then the empty set is a subset of $S$. What are $\inf$ and $\sup$ of such set? To be honest I don't know what it could be. I'm not talking about of real numbers, but any ...
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### Relation between Supremum and limit superior

If $\sup\limits_{n\ge 1} a_n<\infty$ then obtains that $\limsup\limits_{n\to\infty} a_n<\infty.$ Can you explain that?
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### Is there a meaning to the notation “\arg \sup”?

When $f$ is a function on a set $A$, the notation: $\arg\max_{x\in A} f(x)$ denotes the set of elements of $A$ for which $f$ attains its maximum value. This set may be empty, for example, if $f(x)=x$ ...
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### Exchanging supremum and conditional expectation

I've come across a problem which seems similar to this but quite different and can't find a way of going around it. I am working with a continuous process $Y_t$ generating the filtration $(F_t)$ and ...
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### Doob's submartingale theorem

According to Doob's theorem, If $\{X_n,\mathcal{F}_n, n\in \mathbb{N}^* \}$ is a submartingale and $L_1$ - bounded, it means $$\sup\limits_{n\ge 1} \mathbb{E} (|X_n|)<\infty,$$ then $\{X_n\}$ ...
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### How to find the supremum and infimum of this set?

$\{3-\frac{17}{n}:n\in\Bbb{N}\} \subseteq A \subseteq (-\infty,3].$ Prove that A has an upper bound and find $\sup(A)$.
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### Proving the supremum of this set

Suppose we have a subset of reals A and a real number s exists such that for all natural n, s + 1/n is an upper bound for A while s - 1/n is not an upper bound for A. Apparently s = supA but I can't ...
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### $\sup_{x \in \mathbb R} k^2e^{−kx^2}f(x)≤\sup_{x \in \mathbb R} (k+1)^2e^{−(k+1)x^2}f(x)$?

Assuming that $f$ is bounded, continuous, and non-negative, is it true that $$\sup_{x \in \mathbb R} k^2e^{−kx^2}f(x)≤\sup_{x \in \mathbb R} (k+1)^2e^{−(k+1)x^2}f(x)$$ I have a hard time proving this ...
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### Convergence to supremum using Squeeze Theorem and Limit Theorem?

Suppose that $B$ is a nonempty set of $\mathbb{R}$ that is bounded above. Let b = sup($B$). Prove there is a sequence $b_n \in B$ that converges to b. So I know we can use the Squeeze Theorem, and ...
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### Is sup of max, same as max of sup?

Let $\sigma_1, \sigma_2 \dots \sigma_n$ be functions of $\omega \in \mathbb{R}_+$. Is $\sup_{\omega}(\max_{i=1:n} (\sigma_i))$ same as $\max_{i=1:n}( \sup_{\omega}(\sigma_i))$? Could you also please ...
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### Proving $-1$ is the infimum of $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$

Prove: $\inf\{E\}=-1$ for $E=\{\frac{2}{n}+(-1)^n\mid n\in \mathbb{N}\}$ Let assume the contrary , there is $x\in E$ such that $\frac{2}{n}+(-1)^n\leq-1$. Because we are looking at negative ...
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### Which of following inequality holds?

$f(x,y) = _{\theta,\phi}^{{sup}}$ $||e^{i\theta}x - e^{i\phi}y||_{2}^2$ ,where $x,y \in \mathbb{C^2}$ and $\theta,\phi \in \mathbb{R}$ $f(x,y) \leq ||x||^2 + ||y||^2 - 2Re (\langle x,y \rangle )$ ...
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### How do I calculate the variation of a function?

I am trying to understand how to calculate the variation of a function. In this regard, the book that I am reading offers the following definition - V_g([a,b] = sup \sum_{i=0}^n |f(x_{i+1}) - f(...
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### Hausdorff distance on power sets

Consider a general metric space $(S,d)$, with $d$ a $1$-bounded metric, and let $X,Y \subset S$ be two closed subsets of $S$. Notice that $X$ and $Y$ are not compact. Let $\mathcal{P}(X)$ denote the ...
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### Prove that $M(|f|,S)-m(|f|,S)\leq M(f,S)-m(f,S)$

Let $M(f,A)=sup\{f(x):x\in A\subseteq[a,b]\}$ and let $m(f,A)=inf\{f(x):x\in A\subseteq[a,b]\}$. Given that $|f(x_{0})|-|f(y_{0})|\leq |f(x_{0}-f(y_{0})|\leq M(f,S)-m(f,S)$ for $x_0,y_0\in S$, prove ...
I'm having a little trouble with the following exercise: Let $V \subset\mathbb{R}^p$ be a non-empty, closed set and $a \in \mathbb{R}^p$. For $x, y \in \mathbb{R}^p$ we note $d(x,y) = \|x-y\|$. ...