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taking \every Cauchy sequence of real numbers converges" to be the Completeness Axiom, and then proving that R has the LUB Property. Cauchy’s criterion for convergence 1. Proof of Theorem 1 Let fa ngbe a Cauchy sequence. �]����#��dv8�q��KG�AFe� ���4o
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Example 5: The closed unit interval [0;1] … �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$2D3��u4�,e�&(���f.�>1�.��� �R[-�y��҉��p;�e�Ȝ�ނ�'|g� H��Wَ��}��[H ���lgA�����AVS-y�Ҹ)MO��s��R")�2��"�R˩S������oyff��cTn��ƿ��,�����>�����7������ƞ�͇���q�~�]W�]���qS��P���}=7Վ��jſm�����s�x��m�����Œ�rpl�0�[�w��2���u`��&l��/�b����}�WwdK[��gm|��ݦ�Ձ����FW���Ų�u�==\�8/�ͭr�g�st��($U��q�`��A���b�����"���{����'�; 9)�)`�g�C� 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- Theorem 1 Every Cauchy sequence of real numbers converges to a limit.
2 0 obj then completeness will guarantee convergence. 2 MATH 201, APRIL 20, 2020 Homework problems 2.4.1: Show directly from the de nition that ˆ n2 1 n2 0

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