it's negative infinity!

good

"Know that each day that dawns may be your last."

Or rather:

"Believe that each day that shines to you is your last one"

a real numbers computable if it is limit of an effectively converging computable sequence of a retional supremum infimum computable if it is supremum of computable of sequence of a rational numbers

1.common translation=finally be good mortals

2.the leaned people became endeavoured,that was proper to arrange quickly

,the injured became better.

It is the smallest number, s, such that x <= s for any element, x, of the set; and if e is any number, however small, then there is at least one element in the set such that x > (s - e) : that is, (s - e) is not an upper bound.

A join and meet are binary operations on the elements of a POSET, or partially ordered set. A join on a set is defined as the supremum with respect to a partial order on the set, provided it exists. A meet on a set is defined either as the unique infimum with respect to the partial order imposed on the set, if the infimum exists.

In mathematics, the symbol for the highest value is often represented by the infinity symbol (∞), which denotes an unbounded limit. In various contexts, such as statistics or optimization, the highest value can also be denoted using the maximum function, often represented as "max." Additionally, in set theory, the supremum (sup) may be used to indicate the least upper bound of a set.

Two examples of continuous lattices are the lattice of real numbers with the usual order, and the lattice of open sets of a topological space ordered by inclusion. Both of these lattices satisfy the property that any subset with a lower bound has an infimum and any subset with an upper bound has a supremum in the lattice.

-4

It depends on what space your in. If its the supremum norm on a function space then just look for the max of the function. If its the euclidean norm then just takes squares, add, take the square root. Whats more interesting is that its often very hard to compute norms. For instance, even computing the norm of a 2x2 matrix is no easy problem if the matrix isn't diagonalizable. Computing the norm of a given operator on a infinite dimensional Hilbert space is very hard indeed...

Let P = { x0, x1, x2, ..., xn} be a partition of the closed interval [a, b] and f a bounded function defined on that interval. Then: * the upper sum of fwith respect to the partition P is defined as: U(f, P) = cj (xj - xj-1) where cj is the supremum of f(x)in the interval [xj-1, xj]. * the lower sum of f with respect to the partition P is defined as L(f, P) = dj (xj - xj-1) where dj is the infimum of f(x) in the interval [xj-1, xj].

Answer: NO

Explanation:

Let's look at an example to see how this works. A is all rational numbers less than 5. So one element of A might be 1 since that is less than 5 or 1/2, or -1/2, or even 0.

Now if we pick 1/2 or 0, clearly that numbers that are greater than them in the set.

So what we are really asking, is there a largest rational number less than 5.

In a set A, we define the define the supremum to be the smallest real number that is greater than or equal to every number in A. So do rationals have a supremum? That is really the heart of the question.

Now that you understand that, let's state an important finding in math:

If an ordered set A has the property that every nonempty subset of A having an upper bound also has a least upper bound, then A is said to have the least-upper-bound property

In this case if we pick any number very close to 5, we can find another number even closer because the rational numbers are dense in the real numbers.

So the conclusion is that the rational number DO NOT have the least upper bound property.

This means there is no number q that fulfills your criteria.

Let me rephrase it. You mean take a bounded subset of real numbers, S, and find a subset of all the upperbounds of S, say D, such that sup S is not in D?

If I get you right, then yes.

Take D := {a : a = sup s + n, n is natural and n < 4} so the first element is sup s + 1 > sup s, and the remaining two terms are even larger than the first one.

But I think I got you wrong, go through the Completeness Axiom.

That is for any two set A and B such that for all a in A, b in B, a <= b, and they share a maximum of one element, then there exist at least one number x such that a <= x <= b

In particular, A have a supremum, sup A <= x and B have an infimum inf B > = x

sup A <= inf B if they are equal then they must be x.

The previous answer was entirely incorrect. Edward did more than Henry did when it comes to changing religion. Henry broke away from Rome in 1534 when he had the Act of Supremacy passed, which made him Supremum Caput (Supreme Head) of the Church of England. He later dissolved the monasteries in two stages, with two acts of Supression (one in 1536 and the other in 1539). He broke away from Rome for a number of reasons, ranging from the desire for a male heir, which is current wife Catherine of Aragon couldn't provide, to genuine religious reasons, such as his conviction that the marriage to Catherine was not valid anyway, as it says in Leviticus XVIII that any man who lies with his brother's wife shall remain childless. Henry decided that this meant, by sleeping with his brother's widow (Catherine was originally married to Henry's older brother Arthur) he had doomed England to be ruled by a woman after he died. When he broked away from Rome, he was able to grant himself an annulment (NOT A DIVORCE!) Despite all of these movements, Henry was a Catholic, and, when he died, that Act of Six Articles was in place, making England a firmly Catholic country.

When his 9 year old son Edward VI came to the throne when Henry died on 28th January, 1547, his Protector the Duke of Somerset, and Edward himself (Edward was a ridiculously bright child, and definitely influenced religion in his country. Later on in his reign, when he became a teenager, he was directly embroiled in religious debates with fellow reformers and Catholics alike) immediately began to change England into a Protestant country. Edward was a fanatic Protestant. (A Protestant is a Christian who protests against Catholic beliefs). Edward was held in high regard by leading Protestant reformers. Martin Bucer, for example, even stated that Edward was "godly to a marvel". Throughout Edward's reign, churches were emptied of their Catholicism and wealth, chantries were closed, Catholics were persecuted, and the structure of church services and episcopacy itself changed. Transubstantiation was denied, the Eucharist became known as the Lord's Supper, and act upon act was passed. I won't mention them all, because it would probably just confuse you.

So, to say that Edward didn't change the church is ridiculous. I don't think people should answer questions without knowing the full details. Edward changed the church from a Catholic institution to a Protestant one, and, although his reign was tragically short, his work couldn't be undone entirely by his half-sister Mary I (Bloody Mary), and, when she died, Elizabeth I revived her half-brother's more moderate laws and died, leaving England a Protestant country. If I recall correctly, the Pope's excommunication sentence on England that was put in place in 1570 continues until this day.