Math 4381
Review problems #2
1. Determine whether the following sequences converge or diverge. Provide proofs for
your answers.
n2  1
a) xn  2
.
2n  1
b) xn  cos(n)
sin( n)
c) xn 
n
d) x1  1, xn  3  xn1 , n  2
2. Compute the limit of the sequence (if it exists). xn  n 2  n  1  n 2  2 . No proof is
necessary.
1
3. Decide whether the series  converges or not.
n 1 n !
4. TRUE-FALSE. If true provide proof, if false provide a counterexample:
a) If ( xn ) is a convergent sequence, then (| xn |) is also a convergent sequence.
b) If (| xn |) is a convergent sequence, then ( xn ) is also a convergent sequence.
5. Prove that if ( xn ), ( yn ) are convergent sequences with limits x and y respectively,
then zn  axn  byn is convergent to ax  by .(i.e. do the proof, without referring to
known results).
6. Give an example of two sets, A and B, neither of which is open, but for which A  B
is open.
7. Define the sum of two sets A  B  {x : x  a  b, a  A, b  B} . Prove that if A and B
are open, then A  B is open as well.
8. Decide whether the series converge or diverge. Justify your answers.

n3
a)  3
2
n2 n  n

4( n  1)
b) 
n  0 (2 n  1)(2 n  3)
 n

, n  1 is compact. Provide proof for your
9. Decide whether the set A  1,
 n 1

answer.
 n 1

10. Decide whether the set B  1,  k , n  1 is compact. Provide proof for your
 k 1 2

answer.
11. Give an example of an unbounded closed set.
12. Prove that the intersection of two compact sets is a compact set.
13. Is the union of two compact sets a compact set? Provide proof if true and a
counterexample if false.
14. Find an infinite subset of R with no limit points.
15. Construct a subset of R with exactly two limit points.
16. Find a countable subset of R with uncountably many limit points.
17. Does there exist a subset of R with only one interior point?
18. Does there exist a subset of R with only finitely many interior points?