Math 1121 Calculus (II)
Homework 10-2
(Hand in Problem 2(a)(c)(e), 3, 4, 5, 6 )
1. Let {fn} be a sequence of functions on a set A. If {fn} converges uniformly on A, prove that it converges pointwise on A.
2. Determine whether the following sequence of functions {fn} converges pointwise on the indicated interval. If yes, find the function f such that lim
n→∞fn(x) = f (x) and moreover determine whether {fn} converges uniformly to f .
(a) fn(x) = √n
x, on [0, 1].
(b) fn(x) = ex
xn, on (1, ∞).
(c) fn(x) = e−x2
n , on R.
(d) fn(x) = 1
1 + (nx − 1)2, on [0, 1].
(e) fn(x) = nxn(1 − x), on [0, 1].
3. Let {fn} be a sequence of functions defined on R and A1, A2, A3, · · · be sets on R. Prove that
(a) if {fn} converges uniformly to f and the sets A1 and A2, then {fn} converges uni- formly to f on A1∪ A2.
(b) if {fn} converges uniformly to f and the sets A1, A2, · · · , Ak then {fn} converges uniformly to f on A1∪ A2∪ · · · ∪ Ak.
(c) if {fn} converges uniformly to f and the sets Ak for k = 1, 2, · · · , determine whether {fn} converges uniformly to f on ∪∞k=1Ak.
4. Consider the sequence {fn} defined by fn(x) = nx
1 + nx, for x ≥ 0.
(a) Find f (x) = lim
n→∞fn(x).
(b) Show that for 0 < a, {fn} converges uniformly to f on [a, ∞).
(c) Show that {fn} does not converge uniformly to f on [0, ∞).
5. (a) Suppose that {fn} is a sequence of bounded (not necessarily continuous) functions on [a, b] which converges uniformly to f on [a, b]. Prove that f is bounded on [a, b].
(b) Find a sequence of continuous functions on [a, b] which converge pointwise to an unbounded function on [a, b].
6. (Uniform Cauchy criterion ) Prove that let {fn} be a sequence of functions defined on A and be uniformly convergent on A if and only if for > 0, there exists N ∈ N such that for any m, n > N
sup
x∈A
|fn(x) − fm(x)| < .
7. In class, we prove that if {fn} is a sequence of integrable functions defined on [a, b] and converges uniformly to f on [a, b], then f is integrable on [a, b]. But this result is not true if the domains of those functions are not bounded.
(a) Construct a sequence of integrable functions {fn} defined on [0, ∞) and a function f also defined on [0, ∞) such that {fn} converges uniformly to f on [0, ∞) but f is not integrable on [0, ∞).
(b) (Under the same hypothesis of the above problem, even if “f is integrable on [0, ∞)”, this result may still not be true .)
Construct a sequence of integrable functions {fn} defined on [0, ∞) and an integrable function f also defined on [0, ∞) such that {fn} converges uniformly to f on [0, ∞) but
n→∞lim Z ∞
0
fn(x) dx 6=
Z ∞ 0
f (x) dx
