The Real And Complex Number Systems

lim n→
inf a n .
That is,
lim n→
inf a n sup c n .
n
If every c n −, then we define
lim n→
inf a n −.
Remark The concept of lower limit and upper limit first appear in the book (Analyse
Alge’brique) written by Cauchy in 1821. But until 1882, Paul du Bois-Reymond
gave explanations on them, it becomes well-known.
Example 1 −1 n
n1
0,2,0,2,..., sowehave
b n 2andc n 0 for all n
which implies that
lim sup a n 2 and lim inf a n 0.
Example −1 n
n n1
−1, 2, −3,4,..., sowehave
b n and c n − for all n
which implies that
lim sup a n and lim inf a n −.
−n n1
Example −1, −2, −3, . . . , sowehave
b n −n and c n − for all n
which implies that
lim sup a n − and lim inf a n −.
Relations with convergence and divergence for upper (lower) limit
Theorem Let a n be a real sequence, then a n converges if, and only if, the upper
limit and the lower limit are real with
lim n→
sup a n lim n→
inf a n lim n→
a n .
Theorem Let a n be a real sequence, then we have
(1) lim n→ sup a n a n has no upper bound.
(2) lim n→ sup a n − for any M 0, there is a positive integer n 0 such
that as n ≥ n 0 ,wehave
a n ≤−M.
(3) lim n→ sup a n a if, and only if, (a) given any 0, there are infinite
many numbers n such that
a − a n
and (b) given any 0, there is a positive integer n 0 such that as n ≥ n 0 ,wehave
a n a .
Similarly, we also have
Theorem Let a n be a real sequence, then we have