Math 414: Analysis I Homework 6 Due: March 7, 2014
Name :
The following problems are for additional practice and are not to be turned in: (All problems come from Basic Analysis, Lebl )
Exercises: 2.4.4, 2.4.5
Turn in the following problems.
1. Exercise 2.3.10 in Basic Analysis, Lebl
Remark: This gives us another way of proving the Bolzano-Weierstrass Theorem.
Note that this method is a bit weaker than what was done in class, because although it guarantees the convergence of a subsequence, we don’t know exactly what it’s limit could be.
2. Exercise 2.4.1 in Basic Analysis, Lebl
3. Give examples of properly divergent sequences ( x n
) and ( y n
) with y n
= 0 for all n ∈
N such that
(a) ( x n
/y n
) is convergent,
(b) ( x n
/y n
) is properly divergent.
4. Let ( x n
) and ( y n
) be sequences of positive numbers lim( x n
/y n
) = 0.
(a) Show that if lim( x n
) = + ∞ , then lim( y n
) = + ∞ .
(b) Show that if ( y n
) is bounded, then lim( x n
) = 0.
5. Let ( x n
) be properly divergent and let ( y n
) be such that lim( x n y n
) belongs to
R
.
Show that ( y n
) converges to 0.
6. Exercise 2.4.2 in Basic Analysis, Lebl
Remark: A sequence that satisfies the definition above is called a contractive sequence.
7. If x
1
> 0 and x n +1
:=
1
2 + x n
, prove that ( x n
) converges. Find the limit.
Hint: Use the previous exercise.
8. Show directly from definition, that if ( x n
) and ( y n
) are Cauchy sequences, then
( x n y n
) is a Cauchy sequence.
9. Let ( x n that ( x n
) be a Cauchy sequence such that x n
) is ultimately constant. That is, there is some K ∈ for all n ≥ K .
is an integer for every n ∈
N
. Show
N
, such that x n
= a
1