Homework #10
Cauchy Sequences
Read section 2.5
Exercises from the text: Pages 103-106: #1.#2,
Other exercises in thought and logic that you will most certainly find interesting.
1.
Suppose that an  converges to A and that the set an : n
A is an accumulation point of an : n
2.

is infinite. Prove that
.
Let an  be a sequence such that an 1  an 
Cauchy. Does the result hold if an 1  an 
1
for all n 
2n
1
for all n 
n
. Prove that an  is
?
In exercise 3-6 determine whether the given statement is true or false: If true, give an
example, if false prove.
3.
There exists a sequence an  such that the set an : n
points.
 has exactly two accumulation
4.
There exists a sequence an  such that the set an : n
 has exactly m accumulation
points where m .
5.
There exists a sequence an  such that the set of accumulation points of an : n
countably infinite.
 is
6.
There exists a sequence an  such that the set of accumulation points of an : n
uncountably infinite.
 is