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).

The supremum (plural suprema) of a subset $S$ of a partially ordered set $T$ is the least element of $T$ that is greater than or equal to all elements of $S$. It is usually denoted $\sup S$. The term least upper bound (abbreviated as lub or LUB) is also commonly used.

The infimum (plural infima) of a subset $S$ of a partially ordered set $T$ is the greatest element of $T$ that is less than or equal to all elements of $S$. It is usually denoted $\inf S$. The term greatest lower bound (abbreviated as glb or GLB) is also commonly used.

Suprema and infima of sets of real numbers are common special cases that are especially important in analysis. However, the general definitions remain valid in the more abstract setting of order theory where arbitrary partially ordered sets are considered.

2618 questions
51
votes
4 answers

max and min versus sup and inf

What is the difference between max, min and sup, inf?
piotrek
  • 653
  • 1
  • 6
  • 6
46
votes
5 answers

How can I prove $\sup(A+B)=\sup A+\sup B$ if $A+B=\{a+b\mid a\in A, b\in B\}$

If $A,B$ non empty, upper bounded sets and $A+B=\{a+b\mid a\in A, b\in B\}$, how can I prove that $\sup(A+B)=\sup A+\sup B$?
Lona Payne
  • 663
  • 1
  • 8
  • 8
43
votes
3 answers

"sup" in an equation

I am currently reading through JC Lagarias' "The $3x+1$ Problem and its Generalizations" and have come across some notation reading : $$\sup_{K \ge 0} T^{(K)}(N)$$ Now I assume that this means "suppose that $K$ is greater than or equal to $0$",…
Ben Hortin
  • 433
  • 1
  • 4
  • 4
34
votes
5 answers

Proof that $\inf A = -\sup(-A)$

Let $A$ be a nonempty subset of real numbers which is bounded below. Let $-A$ be the set of of all numbers $-x$, where $x$ is in $A$. Prove that $\inf A = -\sup(-A)$ So far this is what I have Let $\alpha=\inf(A)$, which allows us to say that…
25
votes
3 answers

Supremum of all y-coordinates of the Mandelbrot set

Let $M\subset \mathbb R^2$ be the Mandelbrot set. What is $\sup\{ y : (x,y) \in M \}$? Is this known? To be more descriptive: What is the supremum of all y-coordinates of all black points in the following picture: Picture File:Mandel zoom 00…
20
votes
2 answers

supremum of expectation $\le$ expectation of supremum?

Suppose that $X$ is an arbitrary random variable, is the following is true for any function $f$: $$\underset{y\in \mathcal Y} \sup \mathbb E\big[f(X,y)\big] \le \mathbb E\big[\underset{y\in \mathcal Y} \sup f(X,y)\big]?$$ If $f$ is convex in $X$,…
19
votes
4 answers

Sum of the supremum and supremum of a sum

Consider two real-valued functions of $\theta$, $f(\cdot): \Theta \subset\mathbb{R}\rightarrow \mathbb{R}$ and $g(\cdot):\Theta \subset \mathbb{R}\rightarrow \mathbb{R}$. Is there any relation between (1) $\sup_{\theta \in \Theta}…
TEX
  • 192
  • 3
  • 17
  • 55
18
votes
4 answers

How to deal with lim sup and lim inf?

I am currently taking first course in real analysis following Ross's Elementary Analysis textbook. When I was introduced to lim sup and lim inf, I found it hard to manage to play around or make meaningful conclusions from them because the terms are…
user453616
17
votes
2 answers

Proving rigorously the supremum of a set

Suppose $\emptyset \neq A \subset \mathbb{R} $. Let $A = [\,0,2).\,\,$ Prove that $\sup A = 2$ This is my attempt: $A$ is the half open interval $[\,0,2)$ and so all the $x_i \in A$ look like $0 \leq x_i < 2$ so clearly $2$ is an upper bound. To…
16
votes
1 answer

Find $\inf$ and $\sup$ of $a_n = \frac{a_{n-1} + a_{n-2}}{a_{n-3}}$

Sequence $a_n$ is defined in the following way: $a_1 = a_2 = a_3 = 1$ and for $n > 3$: $a_n = \frac{a_{n-1} + a_{n-2}}{a_{n-3}}$. Find $\inf$ and $\sup$ of $A = \{a_n | n \in \mathbb{N}\}$. Edit: turns out finding out $\sup$ and $\inf$ is a hard…
16
votes
3 answers

Least Upper Bound Property Implies Greatest Lower Bound Property

In Rudin $1.11$ Theorem Proof he claims the following Theorem. Suppose $S$ is an ordered set with the least upper bound property $B \subset S$, $B$ is not empty, and $B$ is bounded below. Let $L$ be the set of all lower bounds of $B$. Then …
15
votes
1 answer

Product of sets and supremum

Let $A$ and $B$ be nonempty sets of positive real numbers that are bounded above. Also let $AB = \{ab: a \in A, b \in B \}$. Prove that $AB$ is bounded above and $\sup(AB) = (\sup A) (\sup B)$. So $\sup A$ and $\sup B$ exist by completeness. An…
Damien
  • 4,133
  • 5
  • 29
  • 38
15
votes
1 answer

Polynomial $P(x,y)$ with $\inf_{\mathbb{R}^2} P=0$, but without any point where $P=0$

Recently I've came across such problem: give a polynomial $P(x,y)$, with $\inf_{\mathbb{R}^2} P=0$, but there is no point on the plane where $P=0$. I couldn't solve it after a day, and seriously doubt whether such a function exists, however its…
aplavin
  • 591
  • 1
  • 4
  • 18
14
votes
4 answers

Prove that the sum of the infima is smaller than the infimum of the sum

I'm trying to prove the following inequality: Let $f$ and $g$ be bounded real-valued functions with the same domain. Prove the following: $$\inf(f) + \inf(g) \leqslant \inf(f+g).$$ I thought I had proved it, but I made the erroneous assumption…
user64219
14
votes
3 answers

Show $\inf_f\int_0^1|f'(x)-f(x)|dx=1/e$ for continuously differentiable functions with $f(0)=0$, $f(1)=1$.

Let $C$ be the class of all real-valued continuously differentiable functions $f$ on the interval $[0,1]$ with $f(0)=0$ and $f(1)=1$. How to show that $$\inf_{f\in C}\int_0^1|f'(x)-f(x)|dx=\frac{1}{e}?$$ I have been able to show that $1/e$ is a…
1
2 3
99 100