The Real And Complex Number Systems

Proof: Since g is continuous on a compact disk B (0; M) , g is uniformly
continuous on B (0; M) . Given ε > 0, there exists a δ > 0 such that as
|x − y| < δ, where x, y ∈ S, we have
|g (x) − g (y)| < ε. (*)
For this δ > 0, since f n → f uniformly on S, then there exists a positive
integer N such that as n ≥ N, we have
|f n (x) − f (x)| < δ for all x ∈ S. (**)
Hence, by (*) and (**), we conclude that given ε > 0, there exists a positive
integer N such that as n ≥ N, we have
Hence, h n → h uniformly on S.
|g (f n (x)) − g (f (x))| < ε for all x ∈ S.
9.5 (a) Let f n (x) = 1/ (nx + 1) if 0 < x < 1, n = 1, 2, ... Prove that {f n }
converges pointwise but not uniformly on (0, 1) .
Proof: First, it is clear that lim n→∞ f n (x) = 0 for all x ∈ (0, 1) . Supppos
that {f n } converges uniformly on (0, 1) . Then given ε = 1/2, there exists a
positive integer N such that as n ≥ N, we have
|f n (x) − f (x)| =
1
∣1 + nx∣ < 1/2 for all x ∈ (0, 1) .
So, the inequality holds for all x ∈ (0, 1) . It leads us to get a contradiction
since
1
1 + Nx < 1 2
for all x ∈ (0, 1) ⇒ lim
x→0 + 1
1 + Nx = 1 < 1/2.
That is, {f n } converges NOT uniformly on (0, 1) .
(b) Let g n (x) = x/ (nx + 1) if 0 < x < 1, n = 1, 2, ... Prove that g n → 0
uniformly on (0, 1) .
Proof: First, it is clear that lim n→∞ g n (x) = 0 for all x ∈ (0, 1) . Given
ε > 0, there exists a positive integer N such that as n ≥ N, we have
1/n < ε
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