Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
||
Mirrors > Home > MPE Home > Th. List > df-limsup | Structured version Visualization version Unicode version |
Description: Define the superior limit of an infinite sequence of extended real numbers. Definition 12-4.1 of [Gleason] p. 175. See limsupval 14205 for its value. (Contributed by NM, 26-Oct-2005.) (Revised by AV, 11-Sep-2020.) |
Ref | Expression |
---|---|
df-limsup | inf |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | clsp 14201 | . 2 | |
2 | vx | . . 3 | |
3 | cvv 3200 | . . 3 | |
4 | vk | . . . . . 6 | |
5 | cr 9935 | . . . . . 6 | |
6 | 2 | cv 1482 | . . . . . . . . 9 |
7 | 4 | cv 1482 | . . . . . . . . . 10 |
8 | cpnf 10071 | . . . . . . . . . 10 | |
9 | cico 12177 | . . . . . . . . . 10 | |
10 | 7, 8, 9 | co 6650 | . . . . . . . . 9 |
11 | 6, 10 | cima 5117 | . . . . . . . 8 |
12 | cxr 10073 | . . . . . . . 8 | |
13 | 11, 12 | cin 3573 | . . . . . . 7 |
14 | clt 10074 | . . . . . . 7 | |
15 | 13, 12, 14 | csup 8346 | . . . . . 6 |
16 | 4, 5, 15 | cmpt 4729 | . . . . 5 |
17 | 16 | crn 5115 | . . . 4 |
18 | 17, 12, 14 | cinf 8347 | . . 3 inf |
19 | 2, 3, 18 | cmpt 4729 | . 2 inf |
20 | 1, 19 | wceq 1483 | 1 inf |
Colors of variables: wff setvar class |
This definition is referenced by: limsupcl 14204 limsupval 14205 |
Copyright terms: Public domain | W3C validator |