The Real And Complex Number Systems

if each S n is non-empty by Cantor Intersection Theorem. However (*)
contradicts to g n → 0 pointwise on S. Hence, we know that there exists a
positive integer N such that as n ≥ N,
S n = φ.
That is, given ε > 0, there exists a positive integer N such that as n ≥ N,
we have
|g n (x) − 0| < ε.
So, g n → 0 uniformly on S.
(b) Use he sequence in Exercise 9.5(a) to show that compactness of S is
essential in Dini’s Theorem.
Proof: Let f n (x) = 1 , where x ∈ (0, 1) . Then it is clear that each
1+nx
f n (x) is continuous on (0, 1) , the limit function f (x) = 0 is continuous on
(0, 1) , and f n+1 (x) ≤ f n (x) for all x ∈ (0, 1) . However, f n → f not uniformly
on (0, 1) by Exercise 9.5 (a). Hence, compactness of S is essential in Dini’s
Theorem.
9.10 Let f n (x) = n c x (1 − x 2 ) n for x real and n ≥ 1. Prove that {f n }
converges pointwsie on [0, 1] for every real c. Determine those c for which the
convergence is uniform on [0, 1] and those for which term-by-term integration
on [0, 1] leads to a correct result.
Proof: It is clear that f n (0) → 0 and f n (1) → 0. Consider x ∈ (0, 1) ,
then |1 − x 2 | := r < 1, then
lim f n (x) = lim n c r n x = 0 for any real c.
n→∞ n→∞
Hence, f n → 0 pointwise on [0, 1] .
Consider
f n ′ (x) = n ( c 1 − x 2) ( )
n−1 1
(2n − 1)
2n − 1 − x2 ,
then each f n has the absolute maximum at x n = 1 √ 2n−1
.
As c < 1/2, we obtain that
|f n (x)| ≤ |f n (x n )|
(
n c
= √ 1 − 1 ) n
2n − 1 2n − 1
[√ (
= n c− 1 n
2
1 − 1 ) n ]
2n − 1 2n − 1
10
→ 0 as n → ∞. (*)