1550251515-Classical_Complex_Analysis__Gonzalez_

116 Chapter 2

Examples

1. (IR, d) is complete, whereas (Q, d) is not complete. Here d stands for
   the usual Euclidean metric. We are taking as known from real analysis
   that every Cauchy sequence of real numbers converges to a limit in R


2. (C,d) is also complete. To see this, suppose that {zn} is a Cauchy
   sequence of complex numbers, and let Zn= Xn + iYn· Then given c > 0 we

have Jzm - znl < c form > n > N,,, say. This implies that

and IYm -Ynl < c

whenever m > n > N,,. Hence both {xn} and {Yn} are Cauchy sequences
of real numbers, so they are .both convergent. Suppose that Xn --+ x and
Yn --+ y. Then Zn --+ z' = x + iy. In fact,

since each of the terms lxn - xi and IYn - YI can be made less than^1 / 2 e
by taking n large enough.

Theorem 2.26 If a Cauchy sequence {xn} in a metric space (S,d) con-

tains a convergent subsequence' {xkn}, then the sequence {xnLconverges
to the same point. '

Proof Because { xn} is a Cauchy sequence, for any given e > 0 there exists
N,, such that for m > n > N,,, we have

and also

since km ::'.'.: m. Suppose that Xkm --+ x E S. Then there exists N~ such

that m > N~· implies that

Hence form > n > max(N,,,ND,

which shows that Xn --+ x.

Theorem 2.27 A necessary and sufficient sondition for a metric/ space

( S, d) to be complete is that every sequence of closed nested spheres {En}

in S with radii tending to zero has a nonempty intersection. '

A sequence of spheres is said to be nested if-En\.rl C En for each n.