## 3.4. Lim Sup and Lim Inf

### Definition 3.4.1: Lim Sup and Lim Inf

Let be a sequence
of real numbers. Define

and letA_{j}= inf{ a_{j}, a_{j + 1}, a_{j + 2}, ...}

*c = lim (A*. Then_{j})*c*is called the**limit inferior**of the sequence .Let be a sequence of real numbers. Define

and letB_{j}= sup{ a_{j}, a_{j + 1}, a_{j + 2}, ...}

*c = lim (B*. Then_{j})*c*is called the**limit superior**of the sequence .In short, we have:

*lim inf(a*, where_{j}) = lim(A_{j})*A*_{j}= inf{a_{j}, a_{j + 1}, a_{j + 2}, ...}*lim sup(a*, where_{j}) = lim(B_{j})*B*_{j}= sup{ a_{j}, a_{j + 1}, a_{j + 2}, ...}