Proof. Note that where and are continuous functions of for all , so are bounded on the interval . That is, there are constants such that So each of those integrals composing is bounded, the worst case being It is now possible to bound the sum as well: where is a constant not depending on . It follows that finishing the proof of this lemma. Choosing a value of satisfying both lemmas leads to a non-zero integer () added to a vanishingly small quantity () being equal to zero, is an impossibility. It follows that the original assumption, that can satisfy a polynomial equation with integer coefficients, is also impossible; that is, is transcendental. The transcendence of A similar strategy, different from Lindemann's original approach, can be used to show that the number is transcendental. Besides the gamma-function and some estimates as in the proof for , facts about symmetric polynomials play a vital role in the proof. For detailed information concerning the proofs of the transcendence of and , see the references and external links. See also Transcendental number theory, the study of questions related to transcendental numbers Gelfond–Schneider theorem Diophantine approximation Periods, a set of numbers (including both transcendental and algebraic numbers) which may be defined by integral equations. Notes References External links 602440|Transcendental number (mathematics)} Proof that e is transcendental Proof that the Liouville Constant is transcendental Proof that e is transcendental (PDF) http://www.mathematik.uni-muenchen.de/~fritsch/pi.pdf Articles containing proofs