Math 3210-1
HW 11
Due Friday, July 6, 2007
Monotone and Cauchy Sequences
1. Which of the following sequences are nondecreasing? nonincreasing? bounded? No proofs required.
(a)
(b)
(c)
(d)
(e)
(f)
1
n
(−1)n
n2
5
n
nπ sin
7
(−2)n
n
3n
2. Let (sn ) be a nondecreasing sequence of positive numbers and define σn =
that (σn ) is a nondecreasing sequence.
3. Let s1 = 1 and sn+1 =
s1 + s2 + · · · + sn
. Prove
n
sn + 1
for n ≥ 1.
3
(a) Find s2 , s3 and s4 .
(b) Use induction to show that sn >
1
2
for all n ∈ N.
(c) Show that (sn ) is a nonincreasing sequence.
(d) Show that lim sn exists and find lim sn .
4. Find an example of a sequence of real numbers satisfying each set of properties.
(a) Cauchy, but not monotone
(b) Monotone, but not Cauchy
(c) Bounded, but not Cauchy
Subsequences
5. For each sequence, find the set S of subsequential limits, the lim sup, and the lim inf. No proofs are
needed.
(−1)n
n
(b) (xn ) = (0, 1, 2, 0, 1, 3, 0, 1, 4, . . .)
(a) wn =
(c) yn = n[2 + (−1)n ]
(d) zn = (−n)n
6. Let (rn ) be an enumeration of the set Q. Show that there exists a subsequence (rnk ) such that
limk→∞ rnk = +∞.
Limits of Functions
x
. Determine, by inspection, the limits limx→∞ f (x), limx→0+ f (x),
7. Sketch the function f (x) = |x|
limx→0− f (x), limx→−∞ f (x), and limx→0 f (x) when they exist. Also indicate when they do not exist.
8. Find the following limits and prove your answers.
(a) lim |x|
x→0
x2
x→0 |x|
(b) lim
9. Let f, g and h be functions from D into R, and let c be an accumulation point of D. If f (x) ≤ g(x) ≤
h(x) for all x ∈ D with x 6= c, and if limx→c f (x) = limx→c h(x) = L, then prove that limx→c g(x) = L.
(Note: you must first prove that the limit exists, and then prove it is L.)