The Real And Complex Number Systems

where M = max (M (1) , ..., M (n 0 − 1) , M (n 0 ) + 1) .
9.2 Define two sequences {f n } and {g n } as follows:
(
f n (x) = x 1 + 1 )
if x ∈ R, n = 1, 2, ...,
n
g n (x) =
{ 1
n
Let h n (x) = f n (x) g n (x) .
if x = 0 or if x is irrational,
b + 1 n if x is rational, say x = a b , b > 0.
(a) Prove that both {f n } and {g n } converges uniformly on every bounded
interval.
and
Proof: Note that it is clear that
lim f n (x) = f (x) = x, for all x ∈ R
n→∞
{ 0 if x = 0 or if x is irrational,
lim g n (x) = g (x) =
n→∞ b if x is ratonal, say x = a, b > 0.
b
In addition, in order to show that {f n } and {g n } converges uniformly
on every bounded interval, it suffices to consider the case of any compact
interval [−M, M] , M > 0.
Given ε > 0, there exists a positive integer N such that as n ≥ N, we
have
M
n < ε and 1 n < ε.
Hence, for this ε, we have as n ≥ N
|f n (x) − f (x)| = ∣ x ∣ ≤ M < ε for all x ∈ [−M, M]
n n
and
|g n (x) − g (x)| ≤ 1 n
< ε for all x ∈ [−M, M] .
That is, we have proved that {f n } and {g n } converges uniformly on every
bounded interval.
Remark: In the proof, we use the easy result directly from definition
of uniform convergence as follows. If f n → f uniformly on S, then f n → f
uniformly on T for every subset T of S.
2