fun facts about red winged blackbirds

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(ii) Let x 1 = 1 and for n2N, let x n+1 = f(x n). » 888.9 888.9 888.9 888.9 666.7 875 875 875 875 611.1 611.1 833.3 1111.1 472.2 555.6 With more than 2,400 courses available, OCW is delivering on the promise of open sharing of knowledge. /FontDescriptor 44 0 R endobj /Type/Font 0 0 0 0 0 0 691.7 958.3 894.4 805.6 766.7 900 830.6 894.4 830.6 894.4 0 0 830.6 670.8 » Course: Math 461, Spring 2011 Time & Place: TR 3:10–4:30 pm in RKC 101 Instructor: Jim Belk (belk@bard.edu) �`o&|�# �y�G�s �3 �PxE|r覥{c���8�Z5�SAq��1��ú�"��&�(�9�$3�ab�p�P MIT OpenCourseWare is a free & open publication of material from thousands of MIT courses, covering the entire MIT curriculum.. No enrollment or registration. /LastChar 196 /Name/F8 323.4 354.2 600.2 323.4 938.5 631 569.4 631 600.2 446.4 452.6 446.4 631 600.2 815.5 39 0 obj /Widths[1062.5 531.3 531.3 1062.5 1062.5 1062.5 826.4 1062.5 1062.5 649.3 649.3 1062.5 777.8 694.4 666.7 750 722.2 777.8 722.2 777.8 0 0 722.2 583.3 555.6 555.6 833.3 833.3 /BaseFont/IOXHID+CMR8 /Type/Font << 0 0 0 0 722.2 555.6 777.8 666.7 444.4 666.7 777.8 777.8 777.8 777.8 222.2 388.9 777.8 277.8 500] /BaseFont/UUYCGY+CMR10 791.7 777.8] Math 432 - Real Analysis II Solutions to Homework due February 13 In class, we learned that the n-th remainder for a smooth function f(x) de ned on some open interval containing 0 is given by R n(x) = f(x) nX 1 k=0 f(k)(0) k! /ProcSet[/PDF/Text/ImageC] 277.8 500 555.6 444.4 555.6 444.4 305.6 500 555.6 277.8 305.6 527.8 277.8 833.3 555.6 �Q����X������:Ų؅���A']�\�f.��:[��ih�V����C"�{.��È�[x�����*�j�$�i��)�8� ��D\ /FirstChar 33 >> 1074.4 936.9 671.5 778.4 462.3 462.3 462.3 1138.9 1138.9 478.2 619.7 502.4 510.5 277.8 305.6 500 500 500 500 500 808.6 444.4 500 722.2 777.8 500 902.8 1013.9 777.8 42 0 obj /FirstChar 33 588.6 544.1 422.8 668.8 677.6 694.6 572.8 519.8 668 592.7 662 526.8 632.9 686.9 713.8 /FontDescriptor 38 0 R << Real Analysis II. This note is an activity-oriented companion to the study of real analysis. /Subtype/Type1 /LastChar 196 x��[msܶ��_q�'j�c�7h��8�u'�d�4�((��Ζ�-��w �w�=���"`�}v����ʪZ\-����_~�=]Z�xy��KYZ�x���?Y2�/����R�U!8{��+^|��G|Ɋg��'c����g���ɒ[����_|��fX���~nJC3|۞r�׫�j�>Y /Widths[719.7 539.7 689.9 950 592.7 439.2 751.4 1138.9 1138.9 1138.9 1138.9 339.3 33 0 obj 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 826.4 1062.5 1062.5 826.4 826.4 x��Z[s۶~�P��i�w�d��uڜi����(�L��ؒG�r�?� A�$H+�}�b� ��]�~{�9�����}�����ǗtQ�B-ίƐ�-2����_�&��N3%i�=I3ZH��9=�� �� o�H�>{u�#y�*�8������P��E�(Q�%�wZ��@�L�<9+���6�4�`?>=�����8ы�Ҝhc�S\ 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 275 500 777.8 777.8 777.8 honours undergraduate-level real analysis sequence at the Univer-sity of California, Los Angeles, in 2003. << /BaseFont/MBVDBO+CMTI10 833.3 1444.4 1277.8 555.6 1111.1 1111.1 1111.1 1111.1 1111.1 944.4 1277.8 555.6 1000 /BaseFont/KSPDWT+CMMI10 /Name/F15 1062.5 1062.5 826.4 288.2 1062.5 708.3 708.3 944.5 944.5 0 0 590.3 590.3 708.3 531.3 36 0 obj 820.5 796.1 695.6 816.7 847.5 605.6 544.6 625.8 612.8 987.8 713.3 668.3 724.7 666.7 /FontDescriptor 35 0 R /BaseFont/GXJTNY+CMMI7 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 777.8 /Font 52 0 R << << >> /BaseFont/GRWRJO+CMSY10 /Subtype/Type1 %���� INTRODUCTION TO REAL ANALYSIS II MATH 4332{BLECHER NOTES 5 (The last theorem used to be Homework 1, due Thursday 22 January. No enrollment or registration. endobj /Widths[1000 500 500 1000 1000 1000 777.8 1000 1000 611.1 611.1 1000 1000 1000 777.8 Massachusetts Institute of Technology. 545.5 825.4 663.6 972.9 795.8 826.4 722.6 826.4 781.6 590.3 767.4 795.8 795.8 1091 As it turns out, the intuition is spot on, in several instances, but in some cases (and this is really why Real Analysis is important at /BaseFont/CRIXZL+CMSY7 There exists a complete ordered eld containing Q as a sub eld, and any two complete << 727.8 813.9 786.1 844.4 786.1 844.4 0 0 786.1 552.8 552.8 319.4 319.4 523.6 302.2 /Type/Font 874 706.4 1027.8 843.3 877 767.9 877 829.4 631 815.5 843.3 843.3 1150.8 843.3 843.3 /LastChar 196 << ����*m�w��HwyY�@X�ƒ�|��W�*�g�~�U�mnh$w�E���*L��Yɭ����`�*�Kw>N�r^/ˀ�3�zj��2-�|��Y� 0Bz����=M�:VN�%�Ac��x�ֽ��Q��|��i,�i3T�og�й���NzĻ���#���;��ۼ@� lFA�����z��]z�C���rS�4�Ͷ��%�\�63z��*{1Y��SL� If (X;‰) is a metric space then (1) the whole space X and the empty set?are both open, (2) the union of any collection of open subsets of X is open, (3) the intersection of any flnite collection of open subsets of X is open.

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