Description: If the limsup is not +oo , then the function is eventually bounded. (Contributed by Glauco Siliprandi, 23-Oct-2021)
Ref | Expression | ||
---|---|---|---|
Hypotheses | limsupubuzmpt.j | |
|
limsupubuzmpt.z | |
||
limsupubuzmpt.b | |
||
limsupubuzmpt.n | |
||
Assertion | limsupubuzmpt | |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | limsupubuzmpt.j | |
|
2 | limsupubuzmpt.z | |
|
3 | limsupubuzmpt.b | |
|
4 | limsupubuzmpt.n | |
|
5 | nfmpt1 | |
|
6 | eqid | |
|
7 | 1 3 6 | fmptdf | |
8 | 5 2 7 4 | limsupubuz | |
9 | 6 | a1i | |
10 | 9 3 | fvmpt2d | |
11 | 10 | breq1d | |
12 | 1 11 | ralbida | |
13 | 12 | rexbidv | |
14 | 8 13 | mpbid | |
15 | breq2 | |
|
16 | 15 | ralbidv | |
17 | 16 | cbvrexvw | |
18 | 14 17 | sylib | |