Math 414: Analysis I
Homework 5
Due: February 28th, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: 2.2.1-2.2.8, 2.3.1-2.3.3, 2.3.5(a), 2.3.8
Turn in the following problems.
1. (a) Let c > 0. Without using the Ratio Test for Sequences, prove that lim cn = 0
n→∞
if c < 1, and that {cn } is unbounded (and hence divergent) if c > 1.
Hint: If c > 1, then write c = 1 + r for some r > 0. Use induction to show
1
that (1 + r)n ≥ 1 + nr. If c < 1, write c = 1+r
where r > 0.
(b) Suppose that
|an | ≤
99
100
n
,
for all n ∈ N and let (bn ) be an arbitrary sequence. Prove that (an sin bn ) is
convergent and find its limit.
(c) Suppose that 0 < a < b. Compute (no need to prove anything here, but show
your work!)
an+1 + bn+1
.
n→∞
an + b n
lim
1/n
2. Let (xn ) be a sequence of positive real numbers such that lim(xn ) = L < 1. Show
that there exists a number r with 0 < r < 1 such that 0 < xn < rn for all sufficiently
large n ∈ N. Use this fact to show that lim(xn ) = 0. Extra Credit: Suppose that
1/n
L = 1. If lim(xn ) = L, then can we conclude anything about lim(xn )? Justify
your answer.
3. Prove or give a counterexample: If (xn ) and (yn ) are two divergent sequences, then
(xn yn ) diverges.
√
√
√
4. Let yn := n + 1 − n for n ∈ N. Show that (yn ) and ( nyn ) converge. Find their
limits.
5. Suppose that (xn ) and (yn ) are convergent sequences. Define the sequence (un ) as
un := max {xn , yn }. Prove that (un ) converge. Hint: Refer to exercise 1.3.2 on page
34, in Basic Analysis, Lebl
6. Exercise 2.3.5(b) from Basic Analysis, Lebl
7. Exercise 2.3.7 from Basic Analysis, Lebl
1