The Real And Complex Number Systems

(Lemma) If {a n } and {b n } are two sequences of complex numbers, define
A n =
n∑
a k .
k=1
Then we have the identity
n∑
n∑
a k b k = A n b n+1 − A k (b k+1 − b k )
(i)
k=1
= A n b 1 +
k=1
n∑
(A n − A k ) (b k+1 − b k ) . (ii)
k=1
Proof: The identity (i) comes from Theorem 8.27. In order to show
(ii), it suffices to consider
b n+1 = b 1 +
n∑
b k+1 − b k .
k=1
9.14 Let f n (x) = x/ (1 + nx 2 ) if x ∈ R, n = 1, 2, ... Find the limit function
f of the sequence {f n } and the limit function g of the sequence {f ′ n} .
(a) Prove that f ′ (x) exists for every x but that f ′ (0) ≠ g (0) . For what
values of x is f ′ (x) = g (x)
Proof: It is easy to show that the limit function f = 0, and by f ′ n (x) =
1−nx 2
(1+nx 2 ) 2 , we have
{ 1 if x = 0
lim f n ′ (x) = g (x) =
n→∞ 0 if x ≠ 0 .
Hence, f ′ (x) exists for every x and f ′ (0) = 0 ≠ g (0) = 1. In addition, it is
clear that as x ≠ 0, we have f ′ (x) = g (x) .
(b) In what subintervals of R does f n → f uniformly
Proof: Note that
1 + nx 2
2
≥ √ n |x|
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