# 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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### Let $f: [a,b] \to \Bbb R$ be continuous. Then $f$ is bounded. Prove using sequence and subsequences. [closed]

Let $f: [a,b] \to \mathbb{R}$ be continuous. Then $f$ is bounded. Let $M:= \sup\{f(x): x \in [a, b]\}$ and $m := \inf\{f(x): x \in [a, b]\}$. Then there exist points $c$ and $d$ in $[a, b]$ such that ...
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### Find an upper bound for $\int_{0}^{1}\cos(x)\frac{x f'(x) - f(x) + f(0)}{x^2}dx$

Let $f : [0, 1] \to \mathbb{R}$ such that $f$, $f'$ and $f''$ are continuous on $[0, 1]$ We want an upper bound for $$\int_{0}^{1}\cos(x)\frac{x f'(x) - f(x) + f(0)}{x^2}dx$$ which is strictly bounded ...
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### finding infimum of the ratio of largest to smallest distance of six points in $\mathbb R^2$

How can we find the infimum of the following set $$\left\{\frac{\max_{i\neq j}d(A_i,A_j)}{\min_{i\neq j}d(A_i,A_j)}:A_1,A_2,\ldots,A_6 \in \mathbb R^2, A_i\neq A_j (\forall i\neq j)\right\}.$$ This ...
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### Find infimum and supremum of the function $f$ defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$

Let $A = \{(x, y, z) \in \mathbb R^3 : x^2 + y^2 + z^2 = 2, xy + yz +zx +1 = 0\}$. Find infimum and supremum of the function $f$defined by the formula $f(x, y, z) = 2x+ 2y-3z$ on the set $A$. I want ...
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### Definition od essential supremum for martingales with many measures

Consider the measurable space $(\Omega,\mathbb{P},\mathcal{F})$ and a filtration $\mathbb{F}=\left\{\mathcal{F}_t\mid\,t\in[0,T]\right\}$. Let $\mathcal{Q}$ be a family of equivalent probability ...
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### Let $S=\{\frac{x+1}{x^2+1}:x\in \mathbb{Q},x>0\}$ Prove that $\sup S=\frac{1+\sqrt{2}}{2}$

Let $S=\{\frac{x+1}{x^2+1}:x\in \mathbb{Q},x>0\}$. Only use the knowledge about supremum and infimum. Prove that $\sup S=\frac{1+\sqrt{2}}{2}$. I know the basic method the find the supremum and ...
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