# What can you say about S if inf s sup s?

## What can you say about S if inf s sup s?

If inf S = supS, then s has to be equal to inf S. This means that S contains only one element.

How do you find sup and inf/of a function?

To find a supremum of one variable function is an easy problem. Assume that you have y = f(x): (a,b) into R, then compute the derivative dy/dx. If dy/dx>0 for all x, then y = f(x) is increasing and the sup at b and the inf at a. If dy/dx<0 for all x, then y = f(x) is decreasing and the sup at a and the inf at b.

How do you prove sup s?

The proof for sup(bS) is similar. Let x ∈ bS. Then x/b ∈ S. So sup S ≥ x/b since sup S is an upper bound for S.

### What is sup of a function?

The supremum (abbreviated sup; plural suprema) of a subset of a partially ordered set is the least element in that is greater than or equal to each element of. if such an element exists. Consequently, the supremum is also referred to as the least upper bound (or LUB).

What is inf sup?

The supremum of a set is its least upper bound and the infimum is its greatest upper bound. Definition 2.2. Suppose that A ⊂ R is a set of real numbers.

Can the supremum be infinity?

Explanations (2) A supremum is a fancy word for the smallest number x such that for some set S with elements a1,a2,…an we have x≥ai for all i. In other words, the supremum is the biggest number in the set. If there is an “Infinite” Supremum, it just means the set goes up to infinity (it has no upper bound).

#### What does inf mean in math?

greatest lower bound
In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is a greatest element in that is less than or equal to each element of. if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.

What is the inf of a function?

In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is a greatest element in that is less than or equal to each element of. if such an element exists.

What is the Supremum of an infinite set?

A supremum is a fancy word for the smallest number x such that for some set S with elements a1,a2,…an we have x≥ai for all i. In other words, the supremum is the biggest number in the set. If there is an “Infinite” Supremum, it just means the set goes up to infinity (it has no upper bound).

## Can the supremum equal then infimum?

Yes, one point sets have the same supremum and infimum (actually the same maximum and minimum). If a set A has more two different elements x

What does the prefix inf mean?

Latin: below; after; later.

What is inf English?

Infinitive (abbreviated INF) is a linguistics term for certain verb forms existing in many languages, most often used as non-finite verbs.

### What is the inf function?

In mathematics, the infimum (abbreviated inf; plural infima) of a subset of a partially ordered set is a greatest element in that is less than or equal to each element of. if such an element exists. Consequently, the term greatest lower bound (abbreviated as GLB) is also commonly used.

What is sup mean in math?

What is sup in math?

#### How do you calculate Supremum and Infimum?

If M ∈ R is an upper bound of A such that M ≤ M′ for every upper bound M′ of A, then M is called the supremum of A, denoted M = sup A. If m ∈ R is a lower bound of A such that m ≥ m′ for every lower bound m′ of A, then m is called the or infimum of A, denoted m = inf A. xk.