# 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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### $\int f^{-1}$ when $f$ contains jump discontinuities

I have several questions regarding the following problem. Calculus by Michael Spivak, Chapter 13, Problem 21 (3rd Edition) begins as follows: 13-21. Suppose that $f$ is increasing. Figure 16 suggests ...
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### supremum and summation Inequality

I am trying to prove an Inequality $$\sup_{k \ge 1} \mid x_k \, \mid ^{q-p} \le \, (\mid \sum_{k \ge 1} \mid x_k \mid ^p) ^{q-p \over p}$$ where $1 \le p \le q < \infty$ How should I proceed
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### Find the supreme and infimum of the following set, $\left \{\dfrac{3n}{\sqrt{1+2n^2}}: n\in \Bbb N\right \}$.

Find the supreme and infimum of the set $$A=\left \{\dfrac{3n}{\sqrt{1+2n^2}}: n\in \Bbb N\right \}$$ We claim that, $\inf A=\sqrt{3}$, y $\sup A=\dfrac{3\sqrt{2}}{2}=\dfrac{3}{\sqrt{2}}.$ Let's prove ...
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### Is C_b((a,b)) a complete metric space? [duplicate]

Let $C_b((a,b))$ be the space of continuous and bounded functions on $(a,b) \subset \mathbb{R}$, let $||f||_\infty := \sup\{|f(x)| : x \in (a, b)\}$. Is ($C_b((a,b)),||\cdot||_\infty)$ a complete ...
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### Letting $\varepsilon \to 0$ in proofs

I've seen proofs regarding supremum and infimum of bounded sets in $\mathbb{R}$ involving an arbitrary $\varepsilon > 0$ and 'hard' inequalities (not necessarily strict, but hard to manipulate ...
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### $\sup(A \cdot B) = \sup A \sup B$

Suppose $A$ and $B$ are bounded and nonempty subsets in $\Bbb{R}$. Define the set $A \cdot B$ as follows $$A \cdot B = \{ab\mid a \in A, b \in B\}$$ I tried proving the statement using inequalities ...
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### Find $x$ such that $(ax)^{bx}>c$, where $a,b,c,x>0$

Let $a>0$, $b>0$, and $c>0$. Let $$x_0\triangleq \inf\{x>0:{\rm for~all~} \bar x>x, (a\bar x)^{b\bar x}>c\}.$$ What is a good estimate (least-conservative estimate) for $x_0?$
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### S is a set of adherent values of a sequence $x_n$

Let a sequence $(x_n)_{n\geq 1}$ be a bounded sequence of real numbers. Define $\mathbf S$ as the adherent set of the sequence. Prove that $\lim \inf\;(x_n)=\inf(S)$. An adherent set $\mathbf S$ is a ...
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I am trying to prove the following theorem: let $f:\mathbb [a,b]\to\mathbb R$ be a Riemann integrable function, and let $g:\mathbb [a,b]\to\mathbb R$ be a bounded function. Prove that: $$\overline{\... 1answer 56 views ### Proving a certain inequality of infimums of upper Riemann integral An assignment in Riemann's integral: Given two bounded function f and g in [a,b], if I proved that for every partition p:$$ U(f+g,p) \leq U(f,p)+U(g,p) $$How can I show that: I don't know how to ... 1answer 38 views ### Show that \sup\ A < \sup\ B \implies \exists b\in B \ |\ b \ \text{is upper bound for} A Show that \sup\ A < \sup\ B \implies \exists b\in B \ |\ b \ \text{is upper bound for} A Let \gamma = \sup\ B - \sup\ A > 0 and \epsilon = \frac{\gamma}{2}> 0 further set b=\sup\ B- \... 1answer 39 views ### What does it mean for a family of functions \{f_n\} to be bounded? \{f_n\} is a family of functions continuous on the interval (0,1). I'm forgetting how to define what it means for \{f_n\} to be bounded and my textbook/google searches aren't providing what I ... 1answer 43 views ### Supremum and Continuity for function Let f be continuous on [a, b]. Define a function g as follows: g(a)=f(a) and, for x in (a, b]$$g(x)=\sup \{f(y): y \text { in }[a, x]\}$$Prove that g is monotone increasing and ... 1answer 13 views ### Supremum over real interval equals supremum over rational 'interval' for continuos functions? I've recently read the following reasoning in a paper: the mapping x\mapsto \sup_{s\in[a,b]} f(sx), \mathbb{R} \to \mathbb{R}, is Borel-measurable, since the supremum over s\in[a,b] equals the ... 0answers 58 views ### Prove that \sup\left (\frac{1}{A}\right )=\frac{1}{\inf A}. Let \varnothing\neq A\subseteq \mathbb R^+, such that \inf A>0, then$$\sup\left (\dfrac{1}{A}\right )=\dfrac{1}{\inf A}.$$My Try: Notice that \inf A\leq a for all a\in A. Then, 1/a\leq 1/... 1answer 20 views ### Show that \inf_n\tau_n<t if and only if \exists n:\tau_n<t Let (\tau_n)_{n\in\mathbb N}\subseteq\overline{\mathbb R},$$\tau:=\inf_{n\in\mathbb N}\tau_n$$and t\in\overline{\mathbb R}. Are we able to show that$$\tau<t\Leftrightarrow\exists n\in\mathbb ...
S = $\{a \in \Bbb{Z} | a < 2+1/2$} Given that a is in the set of all integers ($\mathbb{Z}$), would the supremum of this set be equal to $(2 + 1/2)$? If so, given an upper bound of $(2 + 1/2)$, how ...
### Let $A=\{a_k:k \in \mathbb{N}\}$ be a countable set of strictly positive real numbers such that $\mbox{inf}(A) = 0$. Prove an accumulation point.
Let $A=\{a_k:k \in \mathbb{N}\}$ be a countable set of strictly positive real numbers such that $\mbox{inf}(A) = 0$. Prove that $0$ is an accumulation point of A. Must $0$ be the only accumulation ...