The second proof does not need this because showing that the error term vanishes proves the existence of the limit directly.

Eddington was aware that the existence of black holes was theoretically possible, and also realized that the existence of the limit made their formation possible.

* The term border dispute applies only to the many cases where a limit territory bordering more than one state ( including an enclave in one state such as Nagorno Karabakh ) is claimed by two or more, not the very existence of a whole state challenged ( such as the Republic of China, which the People's Republic of China regards as a defunct and illegitimate entity, with its current jurisdiction of Taiwan claimed by the PRC as its 23rd province ).

* The failure to keep records or the existence of contradictory accounts which could limit the opportunity to rebut serious allegations.

defines a matrix ( the conditions for the existence of the limit are given by the Oseledec theorem ).

On the other hand, the existence of algebraic closures in general is usually proved by ' passing to the limit ' from the splitting field result ; which is therefore proved directly to avoid circular reasoning.

It is not known whether the Generalized Suslin Hypothesis is consistent with the Generalized Continuum Hypothesis ; however, since the combination implies the negation of the square principle at a singular strong limit cardinal — in fact, at all singular cardinals and all regular successor cardinals — it implies that the axiom of determinacy holds in L ( R ) and is believed to imply the existence of an inner model with a superstrong cardinal.

Those wishing to use its existence to support dualism have often taken the position that an epistemic gap — particularly if it is a definite limit on our cognitive abilities — necessarily entails a metaphysical gap.

This presupposes the existence of a scaling limit.

While this may seem like a strange requirement, it is actually necessary to preserve the existence of a thermodynamic limit for such systems.

Bendixson's theorem and the Poincaré – Bendixson theorem predict the absence or existence, respectively, of limit cycles of two-dimensional nonlinear dynamical systems.

If the implication of limit preservation is inverted, such that the existence of limits in the range of a function implies the existence of limits in the domain, then one obtains functions that are limit-reflecting.

The existence of the limit can be proved by the means of Bolzano – Weierstrass theorem in a manner almost identical to the proof of existence of arithmetic – geometric mean.

There are extensions of Fekete's lemma that do not require equation ( 1 ) to hold for all m and n. There are also results that allow one to deduce the rate of convergence to the limit whose existence is stated in Fekete's lemma if some kind of both superadditivity and subadditivity is present.

The existence of such atoms is probably theoretically possible as the upper limit for atomic number is likely Z = 173 due to the light-speed limit, after which assigning electron shells would be nonsensical and elements would only be able to exist as ions, but it is not clear if our technology will ever be enough to synthesise them.

* The existence of the thermodynamic limit for the free energy and spin correlations were proved by Ginibre, extending to this case the Griffiths inequality.

Allen was aided by the existence of the six-year limit between such events ; opponents of a state income tax were aware that if a convention were to be held and limited to the property tax issue, another one at which an income tax might be proposed could not be held for at least another six years.

The allowable size of developmental rosters has varied throughout its existence: in the 2005 season, it grew to ten from a former limit of six, but for the 2009 season, it was reduced to a limit of four players.

The existence of such atoms is probably theoretically possible as the upper limit for atomic number is likely Z = 173 due to the light-speed limit, after which assigning electron shells would be nonsensical and elements would only be able to exist as ions, but it is not clear if our technology will ever be enough to synthesise them.

The content is not the same, however: rather than individual security, it is the security and continuing existence of an `` ideological group '' -- those in the `` free world '' -- that is basic.

( Since the time-span of the nation-state coincides roughly with the separate existence of the United States as an independent entity, it is perhaps natural for Americans to think of the nation as representative of the highest form of order, something permanent and unchanging.

This bold self-assertion, after decades of humble subservience, is indeed a twentieth-century phenomenon, an abrupt change in the Southern way of existence.

When Heidegger and Sartre speak of a contrast between being and existence, they may be right, I don't know, but their language is too philosophical for me.

This is the rhetoric of righteousness the beatniks use in defending their way of life, their search for wholeness, though their actual existence fails to reach these `` religious '' heights.

But the highroad, according to the description of its traffic, belongs to life as it is lived in unawareness of death, while the way to the churchyard belongs to some other sort of life: a suffering form, an existence wholly comprised in the awareness of death.

As long as perception is seen as composed only of isolated sense data, most of the quality and interconnectedness of existence loses its objectivity, becomes an invention of consciousness, and the result is a philosophical scepticism.

Whitehead contends that the human way of understanding existence as a unity of interlocking and interdependent processes which constitute each other and which cause each other to be and not to be is possible only because the basic form of such an understanding, for all its vagueness and tendency to mistake the detail, is initially given in the way man feels the world.

As he informs Watson, `` My life is spent in one long effort to escape from the commonplaces of existence.

The monitoring is the highest and most restrictive of any organization in existence.

Thus, it is no mystical intuition, but an analyzable conception to say that man and his tradition can `` fall out of existence ''.

An existentialist is a man who perceives himself only as `` esse '', as existence without substance.

Such a response, of course, misses the point that in crisis order is going out of existence.

But this truth is distorted by its extreme application: the assumption of the separate existence of tradition.

Thus, circular motion is itself one of the essential characteristics of completely perfect celestial existence.

Men seem almost universally to want a sense of function, that is, a feeling that their existence makes a difference to someone, living or unborn, close and immediate or generalized.

My argument is that there was no Saxon Shore prior to that time even though the forts had been in existence since the time of Carausius.

The existence of a community is a state of mind -- a conviction that goals and values are widely shared, that effective communication is possible, that mutual trust is reasonably assured.

This is not to deny the existence of pogroms and ghettos, but only to assert that these horrors have had an effect on the nerves of people who did not experience them, that among the various side effects is the local hysteria of Jewish writers and intellectuals who cry out from confusion, which they call oppression and pain.

It is also consistent with ZF + DC that every set of reals is Lebesgue measurable ; however, this consistency result, due to Robert M. Solovay, cannot be proved in ZFC itself, but requires a mild large cardinal assumption ( the existence of an inaccessible cardinal ).

The most striking feature is the existence of two great lines of depression, due largely to the subsidence of whole segments of the Earth's crust, the lowest parts of which are occupied by vast lakes.

In particular, it is pointed out that he who deliberately kills an animal will on his part be eaten by an animal in a future existence due to karmic retribution.

One fringe theory, supported by paranormal investigator Jon-Erik Beckjord, theorizes that the lack of hard evidence supporting Bigfoot's existence may be due to the creature being an interdimensional being that slips in and out of dimensions.

Phantom cats ( an example of living animals supposedly found outside of their normal range ) are a common subject of cryptozoological interest, largely due to the relative likelihood of existence in comparison to fantastical cryptids lacking any evidence of existence, such as Mothman.

showed that the earlier version of compactness due to Fréchet, now called ( relative ) sequential compactness, under appropriate conditions followed from the version of compactness that was formulated in terms of the existence of finite subcovers.

Because of his special status due to the virgin birth and his pure, unselfish nature, Jesus voluntarily faced his struggle in Gethsemane, death, resurrection, and ascension to show humanity that no phase of mortal existence was beyond God's redeeming love.

Thus, he reasoned that existence must be due to an agent cause that necessitates, imparts, gives, or adds existence to an essence.

The resulting free space, is commonly considered a security risk due to the existence of file recovery software.

* This is the deepest, most subtle level of dukkha ; it includes " a basic unsatisfactoriness pervading all existence, all forms of life, due to the fact that all forms of life are changing, impermanent and without any inner core or substance.

Proof for the existence of a common Germanic goddess once known as * Fraujon does not exist, but scholars have commented that this may simply be due to lack of evidence.

" Although such similarities have given rise to arguments for the existence of a standardized system of urban layout and planning, the similarities are largely due to the presence of a semi-orthogonal type of civic layout, and a comparison of the layouts of Mohenjo-daro and Harappa shows that they are in fact, arranged in a quite dissimilar fashion.

The existence and success of companies using insurance agents is likely due to improved and personalized service.

The listing below incorporates both currently existing institutions, defunct institutions that no longer exist, institutions which never came into existence due to non-ratification of their constitutive instruments, and institutions which do not yet exist, but for which constitutive instruments have been signed.

The existence of an antipope is usually due either to doctrinal controversy within the Church ( heresy ) or to confusion as to who is the legitimate pope at the time ( see schism ).

Jain doctrine states that an object has infinite modes of existence and qualities and they cannot be completely perceived in all its aspects and manifestations, due to inherent limitations of the humans.

In 1619, due to the attacks of Portuguese, independent existence of Jaffna kingdom, came to an end.

Although Somalia has had no central monetary authority for upwards of 15 years between the outbreak of the civil war in 1991 and the subsequent re-establishment of the Central Bank of Somalia in 2009, the nation's payment system is actually fairly advanced due primarily to the widespread existence of private money transfer operators ( MTO ) that have acted as informal banking networks.

But due to the massive post-World War II demobilization of the U. S. armed forces, Kenney's position at the UN Military Staff Committee in New York, and Kenney's unhappiness with being assigned to SAC, for the first two years of its existence, there was some lack of urgency.

In practice, due to the existence of isotopes, molar masses are used instead when calculating the mass ratio.

As an example, a critic of Plantinga's idea of " a mighty nonhuman spirit " causing natural evils may concede that the existence of such a being is not logically impossible but argue that due to lacking scientific evidence for its existence this is very unlikely and thus it is an unconvincing explanation for the presence of natural evils.

A lot of institutions can be said to exist only because the government provides the framework for their existence ; for instance, Marxists argue that the institution of private property only exists due to government.