:: Convergent Real Sequences. Upper and Lower Bound of Sets of Real Numbers
:: by Jaros{\l}aw Kotowicz
::
:: Received November 23, 1989
:: Copyright (c) 1990 Association of Mizar Users
theorem :: SEQ_4:1
canceled;
theorem :: SEQ_4:2
canceled;
theorem :: SEQ_4:3
canceled;
theorem :: SEQ_4:4
canceled;
theorem :: SEQ_4:5
canceled;
theorem :: SEQ_4:6
canceled;
theorem :: SEQ_4:7
canceled;
theorem Th8: :: SEQ_4:8
theorem Th9: :: SEQ_4:9
theorem Th10: :: SEQ_4:10
:: deftheorem Def1 defines bounded_above SEQ_4:def 1 :
:: deftheorem Def2 defines bounded_below SEQ_4:def 2 :
theorem :: SEQ_4:11
canceled;
theorem :: SEQ_4:12
canceled;
theorem :: SEQ_4:13
canceled;
theorem :: SEQ_4:14
theorem :: SEQ_4:15
canceled;
theorem Th16: :: SEQ_4:16
theorem Th17: :: SEQ_4:17
theorem Th18: :: SEQ_4:18
theorem Th19: :: SEQ_4:19
:: deftheorem SEQ_4:def 3 :
canceled;
:: deftheorem Def4 defines upper_bound SEQ_4:def 4 :
:: deftheorem Def5 defines lower_bound SEQ_4:def 5 :
Lm61:
for r being real number
for X being non empty real-membered set st ( for s being real number st s in X holds
s >= r ) & ( for t being real number st ( for s being real number st s in X holds
s >= t ) holds
r >= t ) holds
r = lower_bound X
Lm63:
for X being non empty real-membered set
for r being real number st ( for s being real number st s in X holds
s <= r ) & ( for t being real number st ( for s being real number st s in X holds
s <= t ) holds
r <= t ) holds
r = upper_bound X
theorem :: SEQ_4:20
canceled;
theorem :: SEQ_4:21
canceled;
theorem Th22: :: SEQ_4:22
theorem Th23: :: SEQ_4:23
theorem :: SEQ_4:24
theorem :: SEQ_4:25
theorem Th26: :: SEQ_4:26
theorem :: SEQ_4:27
theorem :: SEQ_4:28
theorem Th29: :: SEQ_4:29
theorem Th30: :: SEQ_4:30
theorem Th31: :: SEQ_4:31
theorem :: SEQ_4:32
theorem Th33: :: SEQ_4:33
theorem :: SEQ_4:34
canceled;
theorem Th35: :: SEQ_4:35
theorem :: SEQ_4:36
theorem Th37: :: SEQ_4:37
theorem :: SEQ_4:38
theorem :: SEQ_4:39
canceled;
theorem Th40: :: SEQ_4:40
theorem :: SEQ_4:41
theorem :: SEQ_4:42
theorem Th43: :: SEQ_4:43
theorem Th44: :: SEQ_4:44
theorem :: SEQ_4:45
theorem :: SEQ_4:46
theorem Th47: :: SEQ_4:47
theorem Th48: :: SEQ_4:48
theorem :: SEQ_4:49
theorem :: SEQ_4:50
theorem :: SEQ_4:51
canceled;
theorem :: SEQ_4:52
canceled;
theorem Th53: :: SEQ_4:53
theorem :: SEQ_4:54
theorem :: SEQ_4:55
theorem Th56: :: SEQ_4:56
theorem Th57: :: SEQ_4:57
theorem :: SEQ_4:58
theorem :: SEQ_4:59
theorem Th60: :: SEQ_4:60
theorem :: SEQ_4:61
theorem Th62: :: SEQ_4:62
theorem :: SEQ_4:63
theorem :: SEQ_4:64
theorem :: SEQ_4:65