Homework #6
Sequences
Read section 2.2
Exercises from the text: Pages 72-74: #2a,g,h, #5, #6, #12, #21
Here are some other exercises in thought and logic that you will most certainly find
interesting.
1. Let A be a bounded subset of with sup( A)  S , Show that there exists a sequence
an with an  A for each n , such that an converges to S.
2. If an converges to a, and bk 
a1  a2  a3   ak
, show that bn also converges
k
to a.
In exercise 3-6 determine whether the given statement is true or false: If true prove, if
false give a counter-example.
3. If an ≤ cn ≤ bn for all n 
and Lim  an   Lim  bn   L, then Lim  cn  exists and is equal
n
n
n
to L.
4. If {an} and {bn} diverge, then {an + bn} diverges.
5. If an  bn for all n , and {an } and {bn } converge to A and B respectively, then A < B.
 3
6. The sequence an    
 5
n
converges.