Math 414: Analysis I
Homework 4
Due: February 19th, 2014
Name:
The following problems are for additional practice and are not to be turned in: (All
problems come from Basic Analysis, Lebl )
Exercises: 2.1.1-2.1.5, 2.1.9-2.1.11, 2.1.13, 2.1.14, 2.1.17, 2.1.18
Turn in the following problems.
1. Show that if (xn ) is unbounded, then there exists a subsequence (xnk ) such that
lim(1/xnk ) = 0.
√
2. Let x1 := 1 and xn+1 := 2 + xn for n ∈ N. Show that (xn ) is convergent and find
the limit. Hint: Consider the Monotone Convergence Theorem here.
3. Suppose that (an ) converges to A, and define a new sequence (bn ) by bn =
for all n ∈ N. Prove that (bn ) converges to A.
an + an+1
2
4. Suppose that (xn ) converges to x0 and (yn ) converges to x0 . Define a sequence (zn )
as follows: z2n = xn and z2n−1 = yn . Prove that (zn ) converges to x0 .
5. Exercise 2.1.6 from Basic Analysis, Lebl
6. Exercise 2.1.7 from Basic Analysis, Lebl
7. Exercise 2.1.12 from Basic Analysis, Lebl
8. Exercise 2.1.15 from Basic Analysis, Lebl
1