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- https://math.libretexts.org/Courses/Coastline_College/Math_C285%3A_Linear_Algebra_and_Diffrential_Equations_(Tran)/10%3A_Some_Prerequisite_Topics/10.02%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/A_First_Journey_Through_Linear_Algebra/09%3A_Appendices/9.04%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Courses/Mission_College/MAT_04C_Linear_Algebra_(Kravets)/08%3A_Some_Prerequisite_Topics/8.02%3A_Well_Ordering_and_Mathematical_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
- https://math.libretexts.org/Courses/Lake_Tahoe_Community_College/A_First_Course_in_Linear_Algebra_(Kuttler)/10%3A_Some_Prerequisite_Topics/10.02%3A_Well_Ordering_and_InductionLet T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, ther...Let T consist of all integers larger than or equal to a which are not in S. The theorem will be proved if T=∅. If T≠∅ then by the well ordering principle, there would have to exist a smallest element of T, denoted as b. It must be the case that b>a since by definition, a∉T. Thus b≥a+1, and so b−1≥a and b−1∉S because if b−1∈ S, then b−1+1=b∈S by the assumed property of S. Therefore, \(b-…
