1549901369-Elements_of_Real_Analysis__Denlinger_

10.

2.6 Subsequences and Cluster Points 115

Find all cluster points of the following sequences. Then use Corollary

2.6.10 or Theorem 2.6.17 to tell whether the sequence converges.

{

(a) 2,^1 1 1 1} {1 1 1 1 }

2

,3,

3

,4,

4

,5,

5

, ... (b) 1,

2

,2,1,

3

,3,1,

4

, 4,1,

5

,5,···

{

(c) 1, -2,^1 1 1 1 }

3

, 4, -5,

6

, 7, -8, g' 10 , -11,

12

, · · ·

(d) {(-1)n5+~} (e) {(-1r(5+~)}

(f) { 5 + (-~r} (g) { sm. n'lf}

6

(h) {cos n

3

7r} (i) {tan

4

:}

11. Prove Theorem 2.6.17 (a).


12. Give an example of a sequence that has exactly 100 cluster points.


13. Show that a sequence can have infinitely many cluster points, by giving
    an example of a sequence that has every natural number as a cluster
    point.

14. Prove that if a monotone sequence { Xn} has a cluster point x (finite or

infinite), then Xn ---+ x.

15. Prove that a sequence that is not bounded above has a subsequence di-

verging to +oo, and a sequence that is not bounded below has a subse-

quence diverging to -oo.

16. Prove that a sequence diverges if and only if it either has more than one
    cluster point or is unbounded.


17. Prove that a sequence { Xn} converges to a real number L if and only if
    every subsequence of { Xn} has a subsequence converging to L.

18. Prove that a sequence {xn} has no convergent subsequence{::} lxnl ---+ oo.

19. Finish the proof of Theorem 2.6. 18 by proving Case 2.


20. Use subsequences to prove that Vx E IR, {sin nx} diverges unless x
    k'lf for some k E N. [Hint: Assume sin nx ---+ L. Case 1: L -:/= 0. Show
    cos nx ---+ ~ by using 2 cos nx sin nx = sin 2nx ---+ L. Take limit of both
    sides of cos 2nx = cos^2 nx - sin^2 nx to get L^2 < 0. Case 2: L = 0. Then
    cos^2 nx = 1 - sin^2 nx ---+ 1. Take the limit of both sides of sin(n + l)x =
    sin n cos x + cos nx sin x to get a contradiction.] Finally, use the above
    results to prove that {cos nx} diverges unless x is an even multiple of 7r.