Sentence examples for proofs of number from inspiring English sources

Unique factorization is essential to many proofs of number theory.

The failure of these seemingly unquestionable principles in turn vitiates the proofs of a number of basic results of classical analysis, for example the Bolzano-Weierstrass theorem, as well as the theorems of monotone convergence, intermediate value, least upper bound, and maximum value for continuous functions[42].

Sometimes, such criteria furnish unified proofs of a number of scattered results in the related literature.

random variables and also obtained very simple proofs of a number of well-known results such as the Hsu-Robbins-Spitzer-Katz theorem.

As the debate concerning the Axiom of Choice rumbled on, it became apparent that the proofs of a number of significant mathematical theorems made essential use of it, thereby leading many mathematicians to treat it as an indispensable tool of their trade.

Over the years several related proofs were published, but the first full proof of Gödel's original result was given only in 1994 by Sam Buss in his 'On Gödel's theorems on lengths of proofs I: Number of lines and speedups for arithmetic.' (Buss 1994).

The proof of the number of equilibria is given in the next theorem.

Know what types of transportation exist in and around the Disney premises, and that, if you stay on-premise and arrive at Orlando International Airport, the Disney Transportation system (bus) can even pick you up and drive you to the intended hotel (with proof of confirmation number of payment, and your Transport will be free).

This project exceeded previous mathematical endeavours by its sheer size, in both length of proof and number of researchers.

But some medical professionals said they are seeing proof of the numbers in their own hospitals.

Current standards for concluding that an agent is a human carcinogen require statistically significant proof of sufficient numbers of cases of cancer in humans with measured or estimated exposures.