Description: A sequence of extended reals converges if and only if its superior limit is smaller than or equal to its inferior limit. (Contributed by Glauco Siliprandi, 2-Dec-2023)
| Ref | Expression | ||
|---|---|---|---|
| Hypotheses | xlimlimsupleliminf.1 | |- ( ph -> M e. ZZ ) |
|
| xlimlimsupleliminf.2 | |- Z = ( ZZ>= ` M ) |
||
| xlimlimsupleliminf.3 | |- ( ph -> F : Z --> RR* ) |
||
| Assertion | xlimlimsupleliminf | |- ( ph -> ( F e. dom ~~>* <-> ( limsup ` F ) <_ ( liminf ` F ) ) ) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | xlimlimsupleliminf.1 | |- ( ph -> M e. ZZ ) |
|
| 2 | xlimlimsupleliminf.2 | |- Z = ( ZZ>= ` M ) |
|
| 3 | xlimlimsupleliminf.3 | |- ( ph -> F : Z --> RR* ) |
|
| 4 | 1 2 3 | xlimliminflimsup | |- ( ph -> ( F e. dom ~~>* <-> ( liminf ` F ) = ( limsup ` F ) ) ) |
| 5 | 1 2 3 | liminfgelimsupuz | |- ( ph -> ( ( limsup ` F ) <_ ( liminf ` F ) <-> ( liminf ` F ) = ( limsup ` F ) ) ) |
| 6 | 4 5 | bitr4d | |- ( ph -> ( F e. dom ~~>* <-> ( limsup ` F ) <_ ( liminf ` F ) ) ) |