Sequences and Series 183
Combining Theorem 2.25 with Example 2 (following Definition 2.42),
we can state the following theorem:
Theorem 4.4 (Cauchy Condition for Convergence). A sequence {zn} in
(C, d) converges iff it is a Cauchy sequence.Thus the condition lzm - Zn I < E Jor both m and n large enough is a
necessary and sufficient condition for convergence. The necessity follows
by specializing Theorem 2.25 to the case of the complex plane, and the
sufficiency was made to depend in Example 2 on the corresponding property
for real sequences. To make our exposition self-contained, we proceed to
establish the sufficiency of the Cauchy condition for real sequences [thus
proving the completeness of (JR, d)]. This we can now do easily in view of
the results in Section 4.4.
Suppose that for any given E > 0 the real sequence { xn} satisfies the
condition lxm - xnl < E whenever m,n >NE. Fixing m = N =[NE]+ 1,
this implies that Ix n I < Ix NI + E for n 2: N, so for all n we have
lxnl:::;: A= max(lx1I, ... , lxN-1J, lxNI + E)
Hence { Xn} is bounded, and by Theorem 4.2, both L1 = lim sup Xn and
L 2 = liminf Xn exist and are finite. Assume that L 1 i= Lz, choose E1 =
(L 1 -L 2 )/3, and let N 1 be the corresponding value of N. By the definitions
of L 1 and Lz we have.andfor some m, n 2: N1. But then
L1 - Lz = (L1 - Xm) + (xm - Xn) + (xn - Lz) < 3e1
contradicting the fact that E 1 = (L 1 - Lz)/3. Therefore, L1 = Lz = L,
and {xn} converges to L.Exercises 4.1
- Let a be a complex number. Determine the convergence or divergence
of the sequence {an}. Find limn--+oo an in the case of convergence - Find limn__, 00 (an - l)/(an + 1) assuming that Jal =f. 1.
3. Show that the sequence Zn = an /(l + a^2 n) converges to zero if either
laJ < 1 or Jal > 1.
- Show that:
(a) lim n (cos !! + i sin !! - 1) = iB
n--+oo n n(
l+·)n
(b) }~n -i = o