Question

Proof that a continuous function on a compact set is uniformly continuous, compact set not containing infimum

score 574 · Answer 1 · 2026-03-29 22:26:41

574

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Proof that a continuous function on a compact set is uniformly continuous, compact set not containing infimum

Published on 29 Mar 2026 - 22:26

score 69 · Answer 2 · 2020-07-03 11:42:14

69

Views

Prove that $ \sup_{s,t\in J}|f\left(t\right)-f\left(s\right)|=\sup_{x\in J}f\left(x\right)-\inf_{x\in J}f\left(x\right) $

Published on 03 Jul 2020 - 11:42

score 57 · Answer 3 · 2020-07-03 13:35:36

$\inf\Big\{\lambda>0:\big|\langle A y,y\rangle\big|\leq\lambda\|y\|^2,\;\forall\,y\in F\Big\}\geq \sup_{\|y\|=1}\big|\langle A y,y\rangle\big|?$

score 81 · Answer 4 · 2026-03-30 22:51:32

81

Views

Prove that $L(f,P)=\inf\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$ and $U(f,P)=\sup\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$

Published on 30 Mar 2026 - 22:51

score 237 · Answer 5 · 2026-03-26 11:05:43

237

Views

How can I prove that this function is bounded?

Published on 26 Mar 2026 - 11:05

score 64 · Answer 6 · 2020-07-05 16:09:18

64

Views

Unable to finish a proof for existence of $n$-th roots

Published on 05 Jul 2020 - 16:09

score 940 · Answer 7 · 2020-07-05 16:51:54

940

Views

How is the $ inf $ defined in a metric space?

Published on 05 Jul 2020 - 16:51

score 56 · Answer 8 · 2020-07-05 22:24:24

56

Views

If $f(x) \leq g(x)$ is it true that $\sup_I(f(x))\leq \sup_I(g(x))$?

Published on 05 Jul 2020 - 22:24

score 258 · Answer 9 · 2020-07-06 13:17:04

258

Views

Is my proof of an upper bound $u$ is the supremum of $\mathit{A}$ iff $\forall(\epsilon>0)$ $\exists a\in\mathit{A}$ such that $u-\epsilon<a$ correct?

Published on 06 Jul 2020 - 13:17

score 99 · Answer 10 · 2020-07-06 15:11:25

99

Views

Confusion about Suprema Properties and Spivak's Proof of the Intermediate Value Theorem

Published on 06 Jul 2020 - 15:11

score 51 · Answer 11 · 2020-07-07 11:19:49

51

Views

Prove: if for all $x,y \in A \subset \Bbb R$, $|x-y|<1$, then $\sup A-\inf A \le 1$

Published on 07 Jul 2020 - 11:19

score 86 · Answer 12 · 2020-07-07 17:40:56

86

Views

$\inf{d(x,y);x∈A, y∈B}>0$

Published on 07 Jul 2020 - 17:40

score 70 · Answer 13 · 2020-07-07 19:34:44

70

Views

Prove: $A,B\in\mathscr P(\Bbb R)\setminus\emptyset ,a \lt b\ \forall a \in A,\forall b \in B\implies \sup A \lt \inf B$

Published on 07 Jul 2020 - 19:34

score 159 · Answer 14 · 2020-07-08 14:04:46

159

Views

Proving that $u=\sup S$ using certain conditions.

Published on 08 Jul 2020 - 14:04

score 214 · Answer 15 · 2020-07-08 16:15:38

214

Views

Infimum of a function over a closed bounded interval

Published on 08 Jul 2020 - 16:15

List Question

Proof that a continuous function on a compact set is uniformly continuous, compact set not containing infimum

Prove that $ \sup_{s,t\in J}|f\left(t\right)-f\left(s\right)|=\sup_{x\in J}f\left(x\right)-\inf_{x\in J}f\left(x\right) $

$\inf\Big\{\lambda>0:\big|\langle A y,y\rangle\big|\leq\lambda\|y\|^2,\;\forall\,y\in F\Big\}\geq \sup_{\|y\|=1}\big|\langle A y,y\rangle\big|?$

Prove that $L(f,P)=\inf\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$ and $U(f,P)=\sup\{\sum_{R\in P} f(x_R)\cdot v(R):x_R\in R\}$

How can I prove that this function is bounded?

Unable to finish a proof for existence of $n$-th roots

How is the $ inf $ defined in a metric space?

If $f(x) \leq g(x)$ is it true that $\sup_I(f(x))\leq \sup_I(g(x))$?

Is my proof of an upper bound $u$ is the supremum of $\mathit{A}$ iff $\forall(\epsilon>0)$ $\exists a\in\mathit{A}$ such that $u-\epsilon<a$ correct?

Confusion about Suprema Properties and Spivak's Proof of the Intermediate Value Theorem

Prove: if for all $x,y \in A \subset \Bbb R$, $|x-y|<1$, then $\sup A-\inf A \le 1$

$\inf{d(x,y);x∈A, y∈B}>0$

Prove: $A,B\in\mathscr P(\Bbb R)\setminus\emptyset ,a \lt b\ \forall a \in A,\forall b \in B\implies \sup A \lt \inf B$

Proving that $u=\sup S$ using certain conditions.

Infimum of a function over a closed bounded interval

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