Double Sequences and Limits

• Autorzy: Noboru Endou , Hiroyuki Okazaki , Yasunari Shidama
• Języki publikacji: EN

Abstrakty

EN

Double sequences are important extension of the ordinary notion of a sequence. In this article we formalized three types of limits of double sequences and the theory of these limits.

Słowa kluczowe

EN

formalization of basic metric space; limits of double sequences

• Wydawca: De Gruyter Open
• Czasopismo: Formalized Mathematics
• Rocznik: 2013
• Tom: 21
• Numer: 3
• Strony: 163-170

Daty

wydano

2013-10-01

Twórcy

autor

Noboru Endou

• Gifu National College of Thechnology Japan

autor

Hiroyuki Okazaki

• Shinshu University Nagano, Japan

autor

Yasunari Shidama

• Shinshu University Nagano, Japan

Bibliografia

• [1] Grzegorz Bancerek. The fundamental properties of natural numbers. Formalized Mathematics, 1(1):41-46, 1990.
• [2] Grzegorz Bancerek. The ordinal numbers. Formalized Mathematics, 1(1):91-96, 1990.
• [3] Czesław Bylinski. Binary operations. Formalized Mathematics, 1(1):175-180, 1990.
• [4] Czesław Bylinski. The complex numbers. Formalized Mathematics, 1(3):507-513, 1990.
• [5] Czesław Bylinski. Binary operations applied to finite sequences. Formalized Mathematics, 1(4):643-649, 1990.
• [6] Czesław Bylinski. Functions and their basic properties. Formalized Mathematics, 1(1): 55-65, 1990.
• [7] Czesław Bylinski. Functions from a set to a set. Formalized Mathematics, 1(1):153-164, 1990.
• [8] Czesław Bylinski. Partial functions. Formalized Mathematics, 1(2):357-367, 1990.
• [9] Czesław Bylinski. Some basic properties of sets. Formalized Mathematics, 1(1):47-53, 1990.
• [10] Noboru Endou, Keiko Narita, and Yasunari Shidama. The Lebesgue monotone convergence theorem. Formalized Mathematics, 16(2):167-175, 2008. doi:10.2478/v10037-008-0023-1.[Crossref]
• [11] Andrzej Kondracki. Basic properties of rational numbers. Formalized Mathematics, 1(5): 841-845, 1990.
• [12] Jarosław Kotowicz. Convergent sequences and the limit of sequences. Formalized Mathematics, 1(2):273-275, 1990.
• [13] Adam Naumowicz. Conjugate sequences, bounded complex sequences and convergent complex sequences. Formalized Mathematics, 6(2):265-268, 1997.
• [14] Jan Popiołek. Some properties of functions modul and signum. Formalized Mathematics, 1(2):263-264, 1990.
• [15] Andrzej Trybulec. Binary operations applied to functions. Formalized Mathematics, 1 (2):329-334, 1990.
• [16] Michał J. Trybulec. Integers. Formalized Mathematics, 1(3):501-505, 1990.
• [17] Zinaida Trybulec. Properties of subsets. Formalized Mathematics, 1(1):67-71, 1990.
• [18] Edmund Woronowicz. Relations and their basic properties. Formalized Mathematics, 1 (1):73-83, 1990.

Identyfikatory

DOI

10.2478/forma-2013-0018