List questions from supremum-and-infimum

Published on 07 Oct 2018 - 21:06

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Prove that $\sup(S-T)=\sup S-\inf T$

Published on 07 Oct 2018 - 23:24

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Any subset of $\mathbb{R}$ is a countable union of pairwise disjoint intervals.

Published on 08 Oct 2018 - 18:51

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Prove that $m$ is a lower bound for $S$ if and only if $−m$ is an upper bound for $−S$.

Published on 10 Oct 2018 - 4:05

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Conjugate function of negative logarithm: fault in my reasoning?

Published on 11 Oct 2018 - 17:14

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Inf, Sup, Min, Max of $\{9+\frac{(-1)^n}{2^n}: n \in \Bbb{N}\}$, $\bigcup_{n=1}^{\infty} [\frac{n+1}{n},3n^2]$

Published on 14 Oct 2018 - 6:00

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Infimum of the set

Published on 15 Oct 2018 - 9:39

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If $\dfrac{x}{y}\in A$ for $x,~y\in A,~x<y$ and $\sup A<1$, prove $\sup A\in A.$

Published on 15 Oct 2018 - 12:00

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Computation of solution of minimization problem

Published on 18 Oct 2018 - 7:37

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Verify proof that $x_n = \sum_{k=1}^n {k \over 2^k}$ is bounded and find its supremum and infinum

Published on 18 Oct 2018 - 17:39

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Prove the limit superior of a bounded sequence converges

Published on 18 Oct 2018 - 22:29

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Continuity of supremum of polynomial

Published on 26 Mar 2026 - 6:07

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Please help me to understand the solution of this problem

Published on 20 Oct 2018 - 12:11

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Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist?

Published on 25 Mar 2026 - 18:47

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Supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$

Published on 21 Oct 2018 - 18:20

List Question

Exercise 7. Chapter 3. Barry Simon.

Prove that $\sup(S-T)=\sup S-\inf T$

Any subset of $\mathbb{R}$ is a countable union of pairwise disjoint intervals.

Prove that $m$ is a lower bound for $S$ if and only if $−m$ is an upper bound for $−S$.

Conjugate function of negative logarithm: fault in my reasoning?

Inf, Sup, Min, Max of $\{9+\frac{(-1)^n}{2^n}: n \in \Bbb{N}\}$, $\bigcup_{n=1}^{\infty} [\frac{n+1}{n},3n^2]$

Infimum of the set

If $\dfrac{x}{y}\in A$ for $x,~y\in A,~x<y$ and $\sup A<1$, prove $\sup A\in A.$

Computation of solution of minimization problem

Verify proof that $x_n = \sum_{k=1}^n {k \over 2^k}$ is bounded and find its supremum and infinum

Prove the limit superior of a bounded sequence converges

Continuity of supremum of polynomial

Please help me to understand the solution of this problem

Does the supremum of the set $S=\{x\in\mathbb{Q}: x>2\}\subset \mathbb{R}$ exist?

Supremum and infimum of $\{x, y \ge 1 : \frac{xy}{3x + 2y + 1}\}$

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