:: RINFSUP2 semantic presentation
theorem Lm021: :: RINFSUP2:1
theorem Lm021x: :: RINFSUP2:2
theorem Lm022: :: RINFSUP2:3
theorem Lm022x: :: RINFSUP2:4
:: deftheorem defines sup RINFSUP2:def 1 :
:: deftheorem defines inf RINFSUP2:def 2 :
:: deftheorem defboundedbelow defines bounded_below RINFSUP2:def 3 :
:: deftheorem defboundedabove defines bounded_above RINFSUP2:def 4 :
:: deftheorem defbounded defines bounded RINFSUP2:def 5 :
theorem LMth36: :: RINFSUP2:5
:: deftheorem Definferior defines inferior_realsequence RINFSUP2:def 6 :
:: deftheorem Defsuperior defines superior_realsequence RINFSUP2:def 7 :
theorem :: RINFSUP2:6
:: deftheorem SEQM3def1 defines increasing RINFSUP2:def 8 :
:: deftheorem SEQM3def2 defines decreasing RINFSUP2:def 9 :
:: deftheorem SEQM3def3 defines non-decreasing RINFSUP2:def 10 :
:: deftheorem SEQM3def4 defines non-increasing RINFSUP2:def 11 :
theorem Lm211: :: RINFSUP2:7
theorem LmXXX: :: RINFSUP2:8
Lm210:
for seq being ExtREAL_sequence holds superior_realsequence seq is non-increasing
Lm220:
for seq being ExtREAL_sequence holds inferior_realsequence seq is non-decreasing
:: deftheorem defines lim_sup RINFSUP2:def 12 :
:: deftheorem defines lim_inf RINFSUP2:def 13 :
theorem Lm202: :: RINFSUP2:9
theorem Lm203: :: RINFSUP2:10
theorem Lm300a: :: RINFSUP2:11
theorem Lm300b: :: RINFSUP2:12
theorem Lm300c: :: RINFSUP2:13
Lm300e:
for seq being ExtREAL_sequence
for rseq being Real_Sequence st seq = rseq holds
( ( rseq is non-increasing implies seq is non-increasing ) & ( seq is non-increasing implies rseq is non-increasing ) & ( rseq is non-decreasing implies seq is non-decreasing ) & ( seq is non-decreasing implies rseq is non-decreasing ) )
theorem Lm300f: :: RINFSUP2:14
theorem Lm300g: :: RINFSUP2:15
theorem Lm3001: :: RINFSUP2:16
theorem Lm3000: :: RINFSUP2:17
theorem Lm3002q: :: RINFSUP2:18
theorem Lm3002y: :: RINFSUP2:19
theorem Lm3002x: :: RINFSUP2:20
theorem :: RINFSUP2:21
theorem :: RINFSUP2:22
theorem Lm3004x: :: RINFSUP2:23
theorem Lm3004: :: RINFSUP2:24
theorem Lm3005: :: RINFSUP2:25
theorem Lm3006: :: RINFSUP2:26
theorem :: RINFSUP2:27
theorem Lm203a: :: RINFSUP2:28
theorem Lm203b: :: RINFSUP2:29
theorem Lm3007: :: RINFSUP2:30
theorem Lm3008: :: RINFSUP2:31
Lm300x:
for seq being ExtREAL_sequence st seq is bounded & seq is non-increasing holds
( seq is convergent_to_finite_number & seq is convergent & lim seq = inf seq )
Lm300y:
for seq being ExtREAL_sequence st seq is bounded & seq is non-decreasing holds
( seq is convergent_to_finite_number & seq is convergent & lim seq = sup seq )
theorem Lm3010: :: RINFSUP2:32
theorem Lm3011: :: RINFSUP2:33
theorem Lm310a1: :: RINFSUP2:34
theorem Lm310a2: :: RINFSUP2:35
theorem Lm310a: :: RINFSUP2:36
theorem Lm310b: :: RINFSUP2:37
theorem LmXX1: :: RINFSUP2:38
theorem :: RINFSUP2:39
Lm330u:
for seq being ExtREAL_sequence st lim_inf seq = lim_sup seq & lim_inf seq = +infty holds
( seq is convergent & lim seq = lim_inf seq & lim seq = lim_sup seq )
Lm330t:
for seq being ExtREAL_sequence st lim_inf seq = lim_sup seq & lim_inf seq = -infty holds
( seq is convergent & lim seq = lim_inf seq & lim seq = lim_sup seq )
Lm330v:
for seq being ExtREAL_sequence st lim_inf seq = lim_sup seq & lim_inf seq in REAL holds
( seq is convergent & lim seq = lim_inf seq & lim seq = lim_sup seq )
theorem Lm330: :: RINFSUP2:40
theorem :: RINFSUP2:41