# 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).

1,582 questions
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### Breaking down $\text{dom}(f)$ to analyze supremum

I am trying to get to the convex conjugate function of the $\ell_1$ norm of a vector $\mathbf{x}$ without using dual norms of indicator functions. My idea so far was: \begin{align*} f^{*}(\mathbf{y})...
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### Why is it necessary to show subsequence convergence in the extreme value theorem?

I’m probably making this more complicated than it needs to be, but I’m trying to figure out why it is necessary to prove the convergence of a subsequence in proving the extreme value theorem (EVT). I ...
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### supremum of additive functions is additive

I need some help for one equality in the following proposition. It was a hint to conclude that $\sup\{f(\cdot):f\in\Phi\}$ is additive. I highlighted it blue. Ultimately I am interested in proving the ...
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### Can't finish this proof about supremums :(

Let $S$ be a nonempty bounded subset of $\Bbb R$ and let $k \in \Bbb R$. Define $kS = \{ks:s \in S\}$. Prove: If $k \ge 0$, then $sup(kS) = k \cdot supS$. Okay, here's my attempt at the proof: ...
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### constructing a saddle function/inf-sup variational statement

I'm trying to make sense of a variational problem under certain conditions. The problem goes as follows: Consider the scalar valued function, $u(\textbf{A},\textbf{b})$, where $\textbf{A}$ is a ...
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### question on the meaning of supremum

I know that supremum means least upper bound. If I have a sequence of events, $\{A_n\}_{n=1}^\infty$ then $$\limsup_{n\rightarrow \infty} A_n = \lim_{n\rightarrow \infty} \sup_{j\geq n} A_j$$ I'm ...
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### Supremum of $M = \{ \left \lfloor{\alpha n} \right \rfloor\frac{1}{n}: n \in \mathbb{N}_{>0}\}$

Lets assume the set $M = \{ \left \lfloor{\alpha n} \right \rfloor\frac{1}{n}: n \in \mathbb{N}_{>0}\}$ with $\alpha \in \mathbb{R}, \alpha > 0$. How can I systematically find the supremum of ...
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### Difference of the sup

Let $f, g:A\to \mathbb R$ two functions. Is it true that $$\sup_{x\in A}|f(x)|-\sup_{x\in A}|g(x)|\leq \sup_{x\in A}|f(x)-g(x)|?$$
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### Is the sup of continuous functions still continuous?

Suppose I have a family of functions $\{f_t, t\in [0, T]\}$, where $f_t:A\to\mathbb R$, with $A$ a generic set (not necessarily contained in $\mathbb R$). Suppose that, for all $t\in [0, T]$, $f_t$ is ...
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### Show that $I=(−\infty,\sup I]$ as $I$ is bounded above but not bounded below

Let $I$ be a non-empty interval. Suppose $I$ is not bounded below, I is bounded above, and $\sup I ∈ I$. Show that $I=(−\infty,c]$, where $c=\sup I$. My attempt:($\Longrightarrow$) Since $I$ is not ...
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### Bounding in Glivenko-Cantelli theorem

Problem Let $X_1, X_2, \cdots, X_n$ be iid random variables. The cdf and empirical cdf are $F(t)=P[X\leq t]$ and $\hat{F}_n(t)=\frac{1}{n} \sum_{i=1}^n 1(X_i\leq t)$. The Glivenko-Cantelli theorem ...
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### $\limsup_{n \to \infty}\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\}\subseteq \{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\leq\epsilon\}$

I recently saw an assertion made that \bigcap_{m \in \mathbb N} \bigcup_{n \geq m}\left\{\frac{X_{n}}{\log{(n)}}\leq\epsilon\right\} \subseteq\left\{\liminf_{n \to \infty}\frac{X_{n}}{\log{(n)}}\...
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### Proving the supremum of a set

Let A be a nonempty subset of $\mathbb R$ and let m be an upper bound of A. Prove that $m = supA$ iff $\forall n \in \mathbb N, \exists a \in A, m < a + \frac 1n$. I don't really have any idea ...
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### Infimum of $n$-th root of $n$

Let $A = \left \{ \sqrt[n]{n} \mid n \in \mathbb{N}\right \}$. I need to find and prove the infimum of $A$. Because $n \in \mathbb{N},$ we can know for sure that $\sqrt[n]{n} \geq 1$. (Does that ...
Please I need help with this question: $E(x)= \left \{ \left(1+ \frac{x}{n} \right)^n : n \in \mathbb{N} \right \}$. Let $a(x) = \sup E(x)$ (least upper bound). ($1$) Prove that $a(x) < a(y)$ if $... 0answers 53 views ### Properties of a supremum of a parametrized set Please I need help solving this question:$E(x)= \left \{ \left(1+ \frac{x}{n} \right)^n : n \in \mathbb{N} \right \}$. Let$a(x) = \sup E(x)$(least upper bound). (1)Prove that$a(x) < a(y)$if$...
Let $A = \left \{ \frac{n-m}{n+m}\mid n,m \in \mathbb{N}, m<n \right \}$ I need to prove the the infimum of $A$ is zero. So I get: \$ \frac{n-m}{n+m} = 1 - \frac{2m}{n+m}\geq 1-\frac{2}{n+1}&...