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* R * x, y * R,z R x 0:⇔∀n N |x|≤ 1 n : x x y :x - y 0 x :⇔∀n N x n x :⇔-x st(x)= z :x z x :⇔∃n N |x|≤ n fin( * R) := {x * R | x } st st * R * R R r R m(r) := {y * R | y r} r r, r1,r2 R

Transcript of R 2 R ) := f 2 R - math.uni-bonn.de · PDF fileN ˆ A) 9h2 N fn2 N j n hg ˆ A

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â Ú m(r) = {r + ε | εØ4Ù'Þ�Ù.ØOà�× Ö2ØOá Üwã }ä Ú m(r) = {y ∈ fin(∗R) | st(y) = r}å Ú r1 6= r2 ⇒ m(r1) ∩m(r2) = ∅ì Ú fin(∗R) =

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²�� �h�4�T���X�¢w�h�H�©�T�∗an

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∃n ∈ N ∀k ∈ ∗N (k ≥ n⇒ |a− ∗

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N − N∗ah ≈ a.

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z �8&[(��e/)�8��*O$��[/!� " �+*W�+/ CVE *W*0�82 �D(>�87K.?�'�∃h ∈ N − N (h ≥ k ∧ |a(k) − a| < ε).

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A�<-¹⇒ º <:z �'&)( ; < 2 " @R7 ∀h, k ∈ ∗N−N ∗ah ≈ ∗ak

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<G �+=�� " �+/0�!� E �T$�5�/V� " � h ∈ ∗N − N

9a := st(∗ah)

< q ≈ " *O7-� E /V� E $ fin(∗R × fin(∗R�%�[=>� " �+/W7:P^���8(��+/D�+� " *O7 " �+/W7 st(∗ah)] y G �'�%��$� @F287����'&)(����4. L 8/0(��+/ " 2 �+� ∀k ∈ ∗N −

N a = st(∗ak)P��'@F*W

∀k ∈ ∗N−N a ≈ ∗ak

<6z �'&)(3���4. z " &[(�70*O7)�8�>�%�8/)�%¼�/ " 70�+/ " E .Ï$�5%/Á /0�4�%N+vX�[/W70�D$�'@F2'7!� " �D½- � L �+/02'�4��ND���+/-�% @F2'��2 �42 �+� a <¹⇐ º <4z �'&)(����4. z " &[(�70*O7)�8�>�%�8/)�%¼�/ " 70�+/ " E .�$�5�/m½t'� L �+/02 �+��N�2 " @R7 ∀h ∈ ∗N−N ∗ah ≈ a

P�'@F*W

∀h, k ∈ ∗N − N (∗ah ≈ a ∧ ∗ak ≈ a)G �

≈� " ����IKJ E "FL �'@F�4�%N+/0�4@M�87 " '� " *O7:P%$�'@M287

∀h, k ∈ ∗N − N ∗ah ≈ ∗ak

E ���3���'&)( ; < � " ��»��+(>� E \%7 E ��2 <

¿©�X�X�¯¢g� �C�2�T�¼�©�,��r� µX�h�¥�g�h�9�����°ªT�ªE³�¦� µX�h�¥�g�h�9���g���w�h�º�[�X�h�X�.� � �X�X�¤ÈkÀ_�h�T�,� É ¡ � ½[« �H�2� �h�¯�R� ���� �X� �,�C�,��X�[�h�O�.��� � �9��[�h����X�[� ±9�2� �h���h�Ê�RÌ�������� Í�� Í��ΦßEÔ ÕrÖºÖ�× Ø

(an)n∈N ∈ RN× Ø4Ù×�ß·×2× ãOã ×?� çwã í[× Ú�Û¼Ü Ù;Ù6ë Ü ú × ÙÓÝhØ�×Rõ çhã í[×�Ù°Ý[× Ùþ¼ß2Ø4à�× ß Øk× Ù µ ý ã à�Ø¥í9ög× ØOà��â Ú +∞

éÙV× Ø¥í[× Ù.à ã Økê2ë.× ß µ ß·× Ùó ø × ß2à�è ç Ù (an)n∈N ∈ RN ⇔ ∀h ∈ ∗N − N∗ah æ;ç Ö·Ø4à�Ø4è&éÙV× ÙVÝ ã Ø�ê2ëä Ú −∞éÙV× Ø¥í[× Ù.à ã Økê2ë.× ß µ ß·× Ùó ø × ß2à�è ç Ù (an)n∈N ∈ RN ⇔ ∀h ∈ ∗N − N

∗ah

Ù×kí Ü à�Ø4è&éÙ× ÙVÝ ã Økê2ëå Ú +∞éÙV× Ø¥í[× Ù.à ã Økê2ë.× ß ¶�· é õ éÙ[íwÖ æ é¦ÙöXà�è ç Ù (an)n∈N ∈ RN ⇔ ∃h ∈ ∗N − N

∗ah æ;ç Ö·Ø4à�Ø4è&éÙV× ÙVÝ ã Ø�ê2ëì Ú −∞éÙV× Ø¥í[× Ù.à ã Økê2ë.× ß ¶�· é õ éÙ[íwÖ æ é¦ÙöXà�è ç Ù (an)n∈N ∈ RN ⇔ ∃h ∈ ∗N − N

∗ah

Ù×kí Ü à�Ø4è&éÙ× ÙVÝ ã Økê2ë��× ø × Ø4Ö Ú ; <-¹

⇒ º <�� � " R ∈ R+<%G �'�%���+� " *O7 " �+/W7-� " � n ∈ N

*W�P>��� Y∀k ∈ N (k ≥ n⇒ ak ≥ R),

�>�8&[(|�^/)�'�%*O$��+/:9∀k ∈ ∗

N (k ≥ n⇒ ∗ak ≥ R).G �� " � @F�4.1�4��70� L '� ∗N − NE ���4���%@ " &)(�2'/0 Y * " �>�HP%2 " @R7Vv " �:�%�[/∀k ∈ ∗

N − N∗ak ≥ R.

ztE �TvS��/t� U �+/R ∈ R+

U �4@ " � U " 2 E �����%�'(%�+/V2 " @R7∀k ∈ ∗

N − N ∀R ∈ R+∗ak ≥ R,

vX'. " 7-� " �D»��4(>� E \�7 E ��2 U �+v " �4*W�4� " *O7 <¹⇐ º < Á �+2 � U �4�|*W� " � " � R ∈ R+

<�z �'&)(�}#'/)� E *W*W�+70N E �%2�2 " @R7∀k ∈ ∗

N − N∗ak ≥ R.

G " ���%'/0.1�4@ÀPp5 U �+/�vX�+@M&)(%�|J E �'��7 " =�N " �+/W7v " /)�HP " *O7v " �:�%�[/ " �670�[/0�eP�� " � C ��vX�+�>� E ��2�%�+*S,t���%�+/Wu�:v�w�/ " ��& " \�@F���%�'(��[/KN E @MQ'*W* " 2 <�� ��+/0(��'@R70�4�Tv " /V� " � n ∈ NP�. " 7�vX�+@M&)(%�4.2 " @R7:9

∀k ∈ ∗N (k ≥ n⇒ ∗ak ≥ R).�^/)�'��*O$��+/-@ " �+$��+/W7

∀k ∈ N (k ≥ n⇒ ak ≥ R),vS�8*VN E N4� " 2'�4�|vX�8/ <A�< Z " /V.5%*W*W�4�3� E /V�%�'*KÉV�4* E @R7)�87V� E * ;'< � E $�� " ���% @F2'� (−an)n∈N

�'��v��4�>���4� <

à

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B6<-¹⇒ º < *K2 " @R7-�>�8&[(�}#8/)� E *W*W�+70N E ��2

∀R ∈ R+∀k ∈ N∃h ∈ N (h ≥ k ∧ ah ≥ R),

vS�8*V* " &[(��>�'&)(|�%�+� U �+/0� " 70*K(>Q E =>2 L �+/WvS�8�>�670�4��ÉV�42 �+@M�37W/)�'�%*O$��+/ " �+/W7-N E∀R ∈ ∗

R+∀k ∈ ∗N∃h ∈ ∗

N (h ≥ k ∧ ∗ah ≥ R).

Z��4���3.?�'�3� E �k,R ∈ ∗N − N

vXQ'(�@R7:P%$�'@F2'7t� " ��»��+(>� E \%7 E ��2 <¹⇐ º <>� � " �4� R ∈ R+, n ∈ N

2 �42'� U �4� <>CVU *W&)(�vSQ8&[(%�4�|���+/V}#8/)� E *W*W�[70N E �%2�@ " �+$��+/W7∃h ∈ ∗

N (h ≥ n ∧ ∗ah ≥ R),

vS�8*V* " &[(�N E∃h ∈ N (h ≥ n ∧ ah ≥ R)7W/)�'�%*O$��+/ " �+/W7 < Á �4�>� E � " �+*-� U �+/KvS��/-N E N4� " 2'�4� <á < *V2'�4�65�287-� " � C ��vX�4��� E ��2�%�+*KÉ-�+* E @R7)�870*V� E * B�< � E $�� " ���% @F2 � (−an)n∈N.

Àj�T�R�>�;� � �X�p�[� � �g�X���9�T��8���[� � �r ∈ ∗R

�h�w�2� �·� µl�y�X� �g� �2� µ ¡ �[�X�h�X�.� � �h�p� �C�'�st(r) := +∞�

st(r) := −∞¡ �,� �,���h�ª/¬&� �©�¦� ���y�h�H�F�h�y� �2� �;�{�h�k�X�[� �y�X�¼���T�/�9�\�X��½[�[«'�h�¥�C�R�h�O½9�;�X�[� �y�&�E���y�h�[���9� ¡¢[�[�9�{µ9�¦�º²>½� ���[�9�w���[�[�g®��y�X�¼��9���[�9�y�h���h��¢w� ���X�h�4� �w�h�¯§V�¥�g�X¢[�w� �,�y�h�Ê

�RÌ�������� Í�� Í��ÎcâVÔ��8ýß(an)n∈N ∈ RN

Ø4Ö·à{a ∈ R | a ¶D· é õ éÙíhÖ æ éÙöXà�è ç Ù (an)n∈N} = {st(∗ah) | h ∈ ∗

N − N}.

��× ø × Ø4Ö Ú G " ��»��4(�� E \�7 E �%2�$�'@F2'7�7)� E 70'@F 2 " *W&)(l� E *����4��wp/0 \� * " 7 " �%�4� A6< A P�ã < B�E �>��ã < á�Pv��4����.?�'��N+v " *W&)(��+��� " 2 �+��70@ " &)(��+� E �>� E �%� " 2 �4��70@ " &)(��4��Ö!Q E $ E ��2'*W\ E ��¼�70�4��� "RÝ �+/0�4�%N " �+/W7 <

À_�X�H�2� �h�R�C£[��»X�;� �w�h���O��k�¼�[� � �h���h��� ½«2�'�[�[�T� �¤�X� ��£[�k� � �h�R��X�X�R�C£[�8­� �T�X�h��;�\�X�[�h�@���[���g�[�9��[�X�� �h���X� �[�h���E� � ���h�h� � �[�9�g�X�_£[¢g�h�R»X�;� �w�h��>�9�T�9��­�9�T�X� ∑∞

i=1 ai

�� �A¿�¢[¢w�\� �.�[� �s : N →

R, n 7→∑n

i=1 ai

�� �E�[« ���;�[�X���/�\� �∗s�[� �y�X�

∗¢w� �����X�h� �y�2�'�E�[�X���y�h�[� � �h«X� � �X��µX�X�O� �T�[¢X�w�k�[�9�w�,�w� ¡��½« ∑k

i=1∗ai

�C�y� �C�s(k)��X�X���C£[�

k ∈ ∗N − N�y�h�[���9�T¢ ��Ê

�RÌ�������� Í�� Í��Î�äÔ�ÕHÖRÖ�× Ø(an)n∈N ∈ RN

éÙVÝa ∈ R ÚEÛAÜ Ù.ÙMíhØ ã à��â Ú ∑∞

n=1 an

ö ç Ù;èh× ßkíhØ�× ß2à8í[×kí[× Ù a⇔ ∀h ∈ ∗N − N∑h

i=1∗ai ≈ aä Ú ∑∞

n=1 an

ö ç Ù;èh× ßkí[× Ù.à ⇔ ∀h, k ∈ ∗N − N∑h

i=1∗ai ≈

∑k

i=1∗ai¬{�[�p® �[�[���¦� �_¹[�\��� ��¢w�T� �;�[�9���� �{®2�;� µX�h�¥�g�X� �y�h�J»X�;� �g�h��>���h� �X�X�_�X�X�h���[�h� ¢w� ���X�X� ¾ � ¡�'� � �h�,�h�¯�9�T�X�A�w���y�h�[� �w�,�y�X�X�|���w�¥�C�y�X� � �[�9� �R� �A��� �h� �C�C�y�[�X�[�w���g�;¢Wå��X®��y�X���h�k� �[�[¢ �'�V�[� �R�9�\�X�¯� �T�X�'�w���¿�¢[¢w�T� �.�g�9�Aµ9�;� ���'�h� � �h���*�h® �,�[�k����[���[�h��® �¦� µX�h�¥�g�h� �,�h�º»h�¦� �g�X�¯�X�h�h�

R��'�E�X�,�y�X�ªe�H� ���y��¿�¢ ¡¢w�T� �.�g�9�¯® ��[�����[�A�� �>�[�h�¯­�[�[� �[�h��¢g� �y�h�[��½�g®��y�X�¯»X�;� �w�h���4�[�T�,�y� �C�'�h�[�;�©� ���F�¥�¦�w�w�·� �2� �;�AÅ;ª  �'� � �g�h���©�\�O�°ª^���h���h�¯»X�[���¼�[� � �h�k�[�9�&�h�k� �9� �h� �y�h�k�R���T���[�9���.�[�O�X�

��Ï���Î0Í�� Í��ÎÄ�EÔºÕrØOÙ×�æ�ú ú�Ø ã ÝhéÙí ` : {a ∈ RN | aú × Ö ê2ëß · Ù¦öXà } → R

ë.× Ø ç/àD»X�'���'&)(�@ " .1�4*'ôõ ÜwãOã Ö©íhØ ã à��â Ú ` Ø4Ö·à æ¦ç Ö·Ø4à�Ø4èä Ú ` Ø4Ö·à ã Ø4Ù× Ü ßå Ú ` ç ßyÝXÙ× à�è9×yÝ[× ß�ö ç Ù.èh× ßkí[× Ù.à�× Ù�ßy×2× ãOã × Ù�� çwã í[×�Ø�ëß·× Ù µ ßy× Ùó ø × ß à8ó�é Ú�0� �C�2�T�¼�©�,���E�[�X�h�X�[� �T�R� �y� ��� �9�,�y�h�¼�2� �h����� �R��9�w�h®w£[�X�� �h���¦�·¸�[� � � � �H��9�w�h¢g�X�Ê

é

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�RÌ�������� Í�� Í��Î�êÔ��8ýßh ∈ ∗N − N

Ö�× à�ó9×Aá Ü Ù`0 : {a ∈ R

N | aú2× Ö�ê2ë[ß · Ùöhà } → R, a 7→ st

(

1

h

h∑

i=1

∗ai

)

`1 : {a ∈ RN | aú2× Ö�ê2ë[ß · Ù¦öXà } → R, a 7→ st(∗ah).

ÛAÜ Ù.ÙMíhØ ã à��â Ú `0 ØOÖ2à�× Ø4Ù èh× ß Ö ê2ëØk×'ú�é¦Ù[íwÖ·Ø4Ù.è Ü ß Ø Ü Ù.à�× ßD� Ü Ù Ü ê2ë ã Ø4á¯× Ö Úä Ú `1 ç ßyÝhÙV×�à>è9×yÝ[× ßAú × Ö ê2ëß · Ù¦öXàC× Ù�� çwã í[×{× Ø4Ù× Ù Ø�ëß·× ß ¶D· é õ éÙíhÖ æ éÙ¦öXà�×Ró�é Ú��× ø × Ø4Ö Ú ; < Z " /Kv�� " *W�4�|�>�8&[(^9

• `0 " *O7V\� * " 7 "FL ] *V2'�4@R70��( " �+/0N E ∀i ∈ N ai ≥ 0< Z " /Kv " *W*W�4���%�'�%�

∀k ∈ N1

k

k∑

i=1

ai ≥ 0,

vS�8*m* " &[(D. " 7m���+/eN E L '/m� E &[(!$�5%/ k ∈ ∗N−N� " ��2'�+$�5%(%/W70�4� z 87)�87 " � ¹ ∑k

i=1∗ai º7W/)�'�%*O$��+/ " �+/W7-N E

∀k ∈ ∗N

1

k

k∑

i=1

∗ai ≥ 0,

�'@F*W " ��* U �4*W'�>���+/0� 1h

∑h

i=1∗ai ≥ 0

E �>�3���8. " 7

st(1

h

h∑

i=1

∗ai) ≥ 0.

• `0 " *O7V@ " ���:��/:] � � " �4� α, β ∈ R< *K2 " @R7

∀k ∈ N1

k

k∑

i=1

(αai + βbi)︸ ︷︷ ︸

αa(i)+βb(i)

= α1

k

k∑

i=1

ai︸︷︷︸

a(i)

+β1

k

k∑

i=1

ai︸︷︷︸

a(i)

.

� " 7t���4��. " 7W70@F�+/WvX� " @F� L �+/W7W/)� E 70�4� � &)(%/0� " U v�� " *W�4�|�+/0(�Q'@R7V.?�'��� E /0&[(��^/)�'�%*O$��+/

∀k ∈ ∗N

1

k

k∑

i=1

(α∗ai + β∗bi)︸ ︷︷ ︸

α∗a(i)+β∗b(i)

= α1

k

k∑

i=1

∗ai︸︷︷︸∗a(i)

+β1

k

k∑

i=1

∗bi︸︷︷︸∗b(i)

,

�'@F*W

∀k ∈ ∗N st(

1

k

k∑

i=1

(α∗ai + β∗bi)︸ ︷︷ ︸

α∗a(i)+β∗b(i)

) = αst(1

k

k∑

i=1

∗ai) + βst(1

k

k∑

i=1

∗bi)

� E $�2'/ E �>���%�[/-Ç " �%�:�8/ " 7)Q87 L � st < � " 7 k = h�[/0(>Q8@F7V.?�'�3*W�9

`0(αa + βb) = α`0(a) + β`0(b).

• `0'/)�����[7 ~ �:���+/1¼f'� L �+/02'�4��70�4�¥/0�4�4@F@F�4�l�% @F2 � " (%/0�4� Á /0�4��N[vX�+/W71N E ] *�2'�4@R70�

limn→+∞ an = a< Z " ��.?�'��� E *!���+/-� " �>� " .1�4�%* " �>�8@F�4� C �>�8@F§�* " *VvX� "RY P�$� @F287����/)� E *!� E &)(

limn→+∞1ai

i=1 ai = aPH���'&)( E �%*W�+/0�4. z " &)(�70*O7)�'������/)�%¼�/ " 70�+¾/ " E .ë� E *�w�/0'\� * " 7 " � ATE ����. " 7 E ��*W�+/0�[/�5 U @ " &[(%�4� z '7)��7 " '��$�5%/����4��Z��+/W7� " ���+/V(�§�\��+/W/0�4�+@M@F�4��ÉV� " (��D�'@F*W 1

h

∑h

i=1∗ai ≈ a

P>�%�'*K(�� "FY 7

st(1

h

h∑

i=1

∗ai) = a.

ì

Page 10: R 2 R ) := f 2 R - math.uni-bonn.de · PDF fileN ˆ A) 9h2 N fn2 N j n hg ˆ A

• `0 " *O7 L �+/0*W&)( " � U�E ��2'* " � L ��/ " �8��7:] � � " ( " �+/0N E a′�%�[=>� " �+/W7!� E /0&)( ∀i ∈ N a

′(i) =a(i + n)

. " 7!� " �%�4. n ∈ N<HG �8����2 " @R7 E ��70�[/ C-E *W� E 70N E ��2T���+/!5 U @ " &)(��+��É-�[¾&[(%�4��/0�42 �4@F��$�5%/V�4�>��@ " &)(�� ��E .1.��4� q � " �D* " &)(�� E /0&)(|2'�4� " 2 �%�+70�4���^/)�'��*O$��+/!� E &[(� E $ C-E *0�6/05�&)¼f�����+/��%'/0. 1

k

∑k

i=1∗bi

. " 7 k ∈ ∗N, b : N → R5 U �[/W7W/)�'2 �+��y

E �������+/ #" 2 �+��*W&)(>��$�70�4� L '� st 9

`0(a′) = st(

1

h

h∑

i=1

ai+n) = st(1

h

h∑

i=1

ai −1

h

n∑

i=1

ai)

= st(1

h

h∑

i=1

∗ai) − st(1

h︸︷︷︸

≈0 (h∈∗N−N!)

n∑

i=1

ai

︸ ︷︷ ︸

∈fin(∗R)︸ ︷︷ ︸

≈0

)

= st(1

h

h∑

i=1

∗ai) − 0 = `0(a).

A�<!G �+/ z �8&[(�v�� " *-�%�[/V� " �%N4�4@F���4� #" 2'�4��*W&)(>��$�70�4���[/W$� @F287t( " �+/V�%�&)(|.�5�(%�4@F *W�+/:9• `1 " *O7V\� * " 7 "FL ] *V2'�4@R70��( " �+/0N E

∀k ∈ N ak ≥ 0,

�>�8&[(|�^/)�'�%*O$��+/∀k ∈ ∗

N∗ak ≥ 0,

" ��* U �4*W ���%�[/0���8@F*W ∗ah ≥ 0.G �

st� " � r /)�%� E ��2 *O/0�+@���7 " �|�[/0(>Q'@R7:9

`1(a) = st(∗ah) ≥ 0.

• `1 " *O7V@ " ���:��/:] � � " �4� α, β ∈ R< *K2 " @R7

∀k ∈ N (αa + βb)(k) = αa(k) + βb(k)︸ ︷︷ ︸

αak+βbk

,

�>�8&[(|�^/)�'�%*O$��+/t�'@F*W∀k ∈ ∗

N∗(αa + βb)(k) = α∗

a(k) + β∗b(k)

︸ ︷︷ ︸

α∗ak+β∗bk

,

" ��* U �4*W ���%�[/0��2 " @F7V. " 7 k = h9

∗(αa + βb)(h) = α∗a(h) + β∗

b(h)︸ ︷︷ ︸

α∗ah+β∗bh

.

� " 7t���+/-Ç " �%�:��/ " 7)Q�7 L � st $�'@M287Vv " � U �4(>� E \�70�+7`1(αa + βb) = α`1(a) + β`1(b).

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