7 4 The lp and l1 spaces 8 1 Vector Spaces 1.1 De nitions A set Xis called a vector space if … Because the Cauchy sequences are the sequences whose terms grow close together, the fields where all Cauchy sequences converge are the fields that are not ``missing" any numbers. h�bbd```b``V �� �q�d�_ f�١`����Tr��`3�c@��/i�H2���A$"��t�������h&�{N�]��8���L/� �� Venkatesha Murthy and B.V. Singbal No part of this book may be reproduced in any form by print, microfilm or any other means with- 0 taking \every Cauchy sequence of real numbers converges" to be the Completeness Axiom, and then proving that R has the LUB Property. ��jj���IR>���eg���ܜ,�̐ML��(��t��G"�O�5���vH s�͎y�]�>��9m��XZ�dݓ.y&����D��dߔ�)�8,�ݾ ��[�\$����wA\ND\���E�_ȴ���(�O�����/[Ze�D�����Z��� d����2y�o�C��tj�4pձ7��m��A9b�S�ҺK2��`>Q`7�-����[#���#�4�K���͊��^hp����{��.[%IC}gh١�? Cauchy sequences are useful because they give rise to the notion of a complete field, which is a field in which every Cauchy sequence converges. 49 0 obj <> endobj The de nition of convergence The sequence xn converges to X when this holds: for any >0 there exists K such that jxn − Xj < for all n K. Informally, this says that as n gets larger and larger the numbers xn get closer and closer to X.Butthe de nition is something you can work with precisely. Proof of Theorem 1 Let fa ngbe a Cauchy sequence. We say that (a n) is a Cauchy sequence if, for all ε > 0 For any j, there is a natural number N j so that whenever n;m N j, we have that ja n a mj 2 j. Remark. /Filter /FlateDecode H�tT�n�0��+t$�H1����� -PE�C���tD�=������ϸ' g��3KR�g��oU��Y��Nf˄�tV�2х�ϓ�"�Ed&3��yA��O�g�� M��a��q2Opy3�@�� �y������E��,a+&&�. %%EOF /Length 4720 Theorem: The normed vector space Rn is a complete metric space. 3.2.3 A sequence in VF that is Cauchy in the l2 norm but not the l1 norm. Theorem 1 Every Cauchy sequence of real numbers converges to a limit. endstream endobj 50 0 obj <> endobj 51 0 obj <> endobj 52 0 obj <>stream 2 0 obj %PDF-1.6 %���� endstream endobj startxref We now consider the sequence fb jggiven by b j = a N j 2 j: Notice that for every nlarger than N stream 9.2 Definition Let (a n) be a sequence [R or C]. �d���v�EP�H��;��nb9�u��m�.��I��66�S��S�f�-�{�����\�1�`(��kq�����"�`*�A��FX��Uϝ�a� ��o�2��*�p�߁�G� ��-!��R�0Q�̹\o�4D�.��g�G�V�e�8��=���eP��L$2D3��u4�,e�&(���f.�>1�.��� �R[-�y��҉��p;�e�Ȝ�ނ�'|g� Remark 354 In theorem 313, we proved that if a sequence converged then it had to be a Cauchy sequence. ю�b�SY`ʀc�����Mѳ:�o� %oڂu�Jt���A�k�#�6� l��.m���&sm2��fD"��@�;D�f�5����@X��t�A�W`�ʥs��(Җ�׵��[S�mE��f��l��6Fιڐe�w�e��,;�V��%e�R3ً�z {��8�|Ú�)�V��p|�҃�t��1ٿ��$�N�U>��ۨX�9����h3�;pfDy���y>��W��DpA 107 0 obj <>stream In fact, as the next theorem will show, there is a stronger result for sequences of real numbers. Cauchy’s criterion for convergence 1. Remark 353 A Cauchy sequence is a sequence for which the terms are even-tually close to each other. then completeness will guarantee convergence. 6���x�����smCE�'3�G������M'3����E����C��n9Ӷ:�7��| �j{������_�+�@�Tzޑ)�㻑n��gә� u��S#��y`�J���o�>�%%�Mw�.��rIF��cH�����jM��ܺ�/�rp��^���0|����b��K��ȿ�A�+�׳�Wv�|DM���Fi�i}RCoU6M���M����>��Rr��X2DmEd��y���]ə >> �]����#��dv8�q��KG�AFe� ���4o ��. Proof: Exercise. We start by rewriting the sequence terms as 2. 2 MATH 201, APRIL 20, 2020 Homework problems 2.4.1: Show directly from the de nition that ˆ n2 1 n2 0/Filter/FlateDecode/ID[<9AAEA8E93178A54115DAC70C20A959B1><514526BA41ED2A49AE780DF13BC0003C>]/Index[49 59]/Info 48 0 R/Length 115/Prev 146218/Root 50 0 R/Size 108/Type/XRef/W[1 3 1]>>stream Example 5: The closed unit interval [0;1] … H��Wَ��}��[H ���lgA�����AVS-y�Ҹ)MO��s��R")�2��"�R˩S������oyff��cTn��ƿ��,�����>�����7������ƞ�͇���q�~�]W�]���qS��P���}=7Վ��jſm�����s�x��m�����Œ�rpl�0�[�w��2���u`��&l��/�b����}�WwdK[��gm|��ݦ�Ձ����FW���Ų�u�==\�8/�ͭr�g�st��($U��q�`��A���b�����"���{����'�; 9)�)`�g�C�

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