Advanced Calculus Assignment 5

1. Advanced Calculus, Assignment 5
1. Let f and g be bounded functions on [a, b]. Prove that
f + g u ≤ f u + g u.
2. (a) Find the pointwise limit of e
x
n
n
n∈N
for x ∈ R.
(b) Is the limit uniform on R?
(c) Is the limit uniform on [0, 1]?
3. Let fn : A → R and f : A → R be functions. Then show that {fn}
converges pointwise to f if and only if for every x ∈ A and for every ε > 0,
there exists N ∈ N such that |fn(x) − f(x)| < ε for all n ≥ N.
4. (a) Suppose {fn} and {gn}, defined on some set A, converge to f and g,
respectively, pointwise. Show that {fn + gn} converges pointwise to
f + g.
(b) Suppose {fn} and {gn}, defined on some set A, converge to f and
g, respectively, uniformly on A. Show that {fn + gn} converges uni-
formly to f + g on A.
5. Suppose there exists a sequence of functions {gn} uniformly converging to
0 on A. Now suppose we have a sequence of functions {fn} and a function
f on A such that
|fn(x) − f(x)| ≤ gn(x)
for all x ∈ A, n ∈ N. Show that {fn} converges uniformly to f on A.
6. Let {fn}, {gn} and {hn} be sequences of functions on [a, b]. Suppose {fn}
and {hn} converge uniformly to some function f : [a, b] → R and suppose
that
fn(x) ≤ gn(x) ≤ hn(x)
for all n ∈ N and for all x ∈ [a, b]. Show that {gn} converges uniformly to
f.
7. Let fn : [0, 1] → R be a sequence of increasing functions (that is, if x ≥ y,
then fn(x) ≥ fn(y) for all n ∈ N). Suppose fn(0) = 0 and limn→∞ fn(1) =
0. Show that {fn} converges uniformly to 0.
8. While uniform convergence preserves continuity, it does not preserve dif-
ferentiability.
(a) Consider fn(x) = |x|1+ 1
n . Show that these functions are differentiable
on [−1, 1], converge pointwise and the limit is not differentiable. Note
that the convergence is in fact uniform (no need to prove).
1

2. (b) Let gn = xn
n . Show that {gn} converges uniformly to a differentiable
function g on [0, 1] (find g). However, show that
g (1) = lim
n→∞
gn (1).
9. (a) Let f : [0, 1] → R be a Riemann integrable (hence bounded) function.
Find limn→∞
1
0
f(x)
n dx.
(b) Show that limn→∞
2
1
e−nx2
dx = 0.
10. Suppose fn : [a, b] → R is a sequence of continuous functions that converge
pointwise to a continuous f : [a, b] → R. Suppose that for any x ∈ [a, b]
the sequence {|fn(x) − f(x)|} is monotone decreasing (with respect to n).
Show that the sequence {fn} converges uniformly.
11. Find a sequence of Riemann integrable functions fn : [0, 1]R such that
{fn} converges to zero pointwise, and such that
(a) {
1
0
fn}∞
n=1 increases without bound.
(b) {
1
0
fn}∞
n=1 is the sequence −1, 1, −1, 1, −1, 1, . . ..
12. Let
h(x) =
∞
n=0
sin(nx)
2n
.
Prove that h(x) < ∞ for all x ∈ R.
13. Let g : (−1, 1) → R be the function g(x) = x
1−x . Show that
∞
n=1 xn
converges pointwise to g, but does not converge uniformly to g on the
open interval (−1, 1).
14. For n = 1, 2, . . ., x ∈ R put fn(x) = x
1+nx2 . Show that {fn} converges
uniformly to a function f and that the equation
f (x) = lim
n→∞
fn (x)
is correct of x = 0, but false if x = 0.
15. Consider the sequence {fn} of functions defined by
fn(x) =
n + cos(nx)
2n + 1
for all x ∈ R.
Show that {fn} is pointwise convergent. Find its pointwise limit.
2