The Real And Complex Number Systems

which implies that
c ≤ a b
since is arbitrary. So,
lim sup a nb n ≤ lim sup a n lim sup b n .
n→ n→ n→
Remark: (1) The equality may NOT hold. For example,
a n 1/n if n is odd and a n 1ifn is even.
and
b n 1ifn is odd and b n 1/n if n is even.
(2) The reader should noted that the finitely many terms does NOT change the relation
of order. The fact is based on the process of the proof.
(3) The reader should be noted that if letting A n log a n and B n log b n , thenby(a)
and log x is an increasing function on 0, , we have proved (b).
8.3 Prove that Theorem 8.3 and 8.4.
(Theorem 8.3) Leta n be a sequence of real numbers. Then we have:
(a) lim inf n→ a n ≤ lim sup n→ a n .
Proof: If lim sup n→ a n , then it is clear. We may assume that
lim sup n→ a n . Hence, a n is bounded above. We consider two cases: (i)
lim sup n→ a n a, wherea is finite and (ii) lim sup n→ a n −.
For case (i), if lim inf n→ a n −, then there is nothing to prove it. We may assume
that lim inf n→ a n a ′ ,wherea ′ is finite. By definition of limit superior and limit inferior,
given 0, there exists a positive integer N such that as n ≥ N, wehave
a ′ − /2 a n a /2
which implies that a ′ ≤ a since is arbitrary.
For case (ii), since lim sup n→ a n −, wehavea n is not bounded below. If
lim inf n→ a n −, then there is nothing to prove it. We may assume that
lim inf n→ a n a ′ ,wherea ′ is finite. By definition of limit inferior, given 0, there
exists a positive integer N such that as n ≥ N, wehave
a ′ − /2 a n
which contradicts that a n is not bounded below.
So, from above results, we have proved it.
(b) The sequence converges if and only if, lim sup n→ a n and lim inf n→ a n are both
finite and equal, in which case lim n→ a n lim inf n→ a n lim sup n→ a n .
Proof: ()Given a n a convergent sequence with limit a. So, given 0, there
exists a positive integer N such that as n ≥ N, wehave
a − a n a .
By definition of limit superior and limit inferior, a lim inf n→ a n lim sup n→ a n .
()By definition of limit superior, given 0, there exists a positive integer N 1 such
that as n ≥ N 1 ,wehave
a n a
and by definition of limit superior, given 0, there exists a positive integer N 2 such that
as n ≥ N 2 ,wehave