converges to A.
Proof (=) If every subsequence of the sequence {an } converges to A, then {an } converges to A, since it is a subsequence of itself.
(=) Suppose that the sequence {an } converges to A. Pick an arbitrary subsequence {ank } and prove that {ank } must also converge to A. Thus, we need to pick an arbitrary c > 0 and look for k* so that
< E
I ank - Al for all nk k'.
Since the sequence {an } converges to A, there exists m1 such that a - Al <J for all n > ml.
Since {ank } is a subsequence {an }, we haven < n2 < , with i < ne, i E N. The equality is true if {ank } is the sequence of itself. Thus, for k ? ml, we have nk ? ml. So pick k* = ml.
Then
tank-AI CE for all nk ? k*.
Therefore, {ank } converges to A. Since {ank } was arbitrary, every subsequence of {an } must converge to A.
Theorem 2.6.5 is very powerful and has many consequences. One is that a sequence {ate } converges to a subsequential limit a of {an } if and only if every subsequence of {an } converges to a. Theorem 2.6.5 can be used to prove divergence. If some subsequence of {an } diverges, then so does {an }; all it takes is one such subsequence. Furthermore, if we can exhibit two subsequences of {an } that converge to two different values, then {an } must diverge. Thus, to prove that the sequence {an }, with a = (- l)' diverges, simply observe that {a2n } converges to 1, {02n i } converges to -1, and both of these are subsequences of {an } . (See Exercise 2(f) from Section 2.1.) In addition, if we know that a sequence converges, we can use subsequences to actually find the limiting value, which is demonstrated in the next example. Recall from previous sections that this task was not always easy.
Example 2.6.6. Consider the sequence {an } with an = " c for any constant c > 0. Prove that {an} converges to 1 (see Exercise 14 in Section 2.1).
Proof If c . = 1, the result is obvious. Why? Next, we prove the convergence to 1 by consider- ing two cases.
Case 1. Suppose that c> 1. If we were to write out a few terms, we would suspect that {an } is decreasing. To prove this, using Remark 2.4.3, part (d), we write
an _ an+1 = _ + c = n+' c n(n+1 c - 1 > 0.
Therefore, {an } is decreasing and clearly bounded below by 1. Thus, using Theorem 2.4.4, {an } converges to, say, A. Subsequences will be used to evaluate A. First, observe that by Theorem 2.2.1, part (e), we have {/} converging to JA. Here, ,./a = " c = ?fc. But
{ 2n c} is a subsequence of (Z/Z}. Thus, { an} must converge to A. Therefore, JA = A. Why?
The two choices for A are 0 and 1. But an > 1, for all n e N, and thus 0 is not a possibility.
Hence, limn 4 " c = 1.
Case 2. A proof for the case 0 < c < 1 is left to Exercise 10.
>
1
-
_ = _ =
Exercises 2.6
1. Determine whether the sequence {bn } is a subsequence of {an }, with an and bn as given.
(a) bn = -1 and an = (-1)'
1 1
(b) bn = and an = -
n
1 1
(c) bn =
3n2 and an - n2
2. Verify that the sequences {an } with an as given diverge. Find all of the subsequential limits, lim supn_ an, and lim infni 0 an .
1 + (-1)n+
(a) an =
nit2 (b) an = sin 2
(c) an = r n, r <-1 or r> 1 (d) a = (_1)n n-1
n n
3. Let the sequence a{ n}be defined by a = 1 + +Y n 231+ `fi1.n (See Exercise 7(1) from(
Section 2.5 and Example 7.1.13.) Prove that {an } is unbounded by showing that there exists some subsequence that is unbounded.
4. Prove that the following two statements are equivalent.
(a) Any bounded sequence must have a converging subsequence (see Theorem 2.6.4).
(b) Every bounded monotone sequence must converge (see Theorem 2.4.4, part (a)).
(Compare this exercise to Exercise 18 from Section 2.4.)
5. Prove that every unbounded sequence contains a monotone subsequence that diverges to infinity.
b. Reprove Theorem 2.5.9 using the Bolzano-Weierstrass theorem for sequences rather than the Bolzano--Weierstrass theorem for sets.
7. Let {an) be a sequence and let set S = {an I n E N } . If so is an accumulation point of S, prove that there exists a subsequence of {an } that converges to so. (Compare to Exercise 4 from Section 2.5.)
8. Suppose that the sequence (an) converges to A, and suppose that B is an accumulation point of the set S = { an I n E N). Prove that A = B.
9. Use subsequences to find the limit of the sequence {an } with an as given.
(a) an = with any constant c E (0, 1). (See Exercise 14 from Section 2.1.)
(b) an = rn with any constant r E (0, 1). (See Exercise 2(c) above and Theo- rem 2.1.13.)
(c) al = 1 and an+1 = tan for all n E N. (See Exercise 11(c) of Section 2.4.) 10. Complete the proof in Example 2.6.6.
11. Complete the proof in Theorem 2.6.4.
,
2
n
'r c,
Sec. 2.7 * Review 111
2.7* Review
Label each statement as true or false. If a statement is true, prove it. If not, (i) give an example of why it is false, and
(ii) if possible, correct it to make it true, and then prove it.
1. If limn. an = A, with A o a constant, then the sequence {an } diverges to infinity.
n
2. The sequence {an }, given recursively by ai = 1 and an+1 = 1 + an for all n E N, converges to the golden number.
3. If the sequence {an } is bounded, then it is Cauchy.
4. If {an } is a bounded sequence with exactly one subsequential limit a, then {an } converges to a.
5. If {an } is a subsequence of {bn } and {bn } is a subsequence of {an }, then {an } = {bn }.
6. If a sequence {an } is bounded and all of its convergent subsequences have the same limit, then {an } converges.
_ n
7. The sequence {a} with a =n n 1) is an oscillating sequence.
n
8. If {an } and {bn } are sequences of positive terms such that limner - an = +oo, and if {an }
bn
is bounded, then (b} converges to zero.
9. The sequence {an } with an = 1 2n
2n is monotone.
14. If the sequence {a } converges to a constant A, then there exists n * such thatn A3 -
a A+ 1 for all n n*.
11. If sequences {an }, {bn }, and {cn } converge to A, B, and AB, respectively, then there exists n` such that for all n ? n, we have cn = an bn .
12. The sequence {an } converges if and only if it is monotone.
13. If any neighborhood of a constant A contains infinitely many terms of a sequence {an }, then {an } converges to A.
14. Suppose that the sequence {an } converges to A. Define a new sequence {bn } by bn =
?a + an+2, for all n E N. Then {b } converges to A.l
3 n 3 n
15. Suppose that {an } and {bn } are two sequences such that {an + bn } converges to A and {a - b} converges to B. Then the sequence {a b } converges ton n n n (A2- B2 .
4
16. If a1 = 3 and an+1 - {(an)2 +61 for all n E N, then the sequence {a } converges to 2.
2 5 n
_
+
17. If limner an = A 0, then the sequence ((- 1)"a,,} diverges.
18. If k is any fixed natural number, then lirnn 0 't nk = I.
19. If k is any fixed natural number, then lirnn 1Z = 1.
aan+ a"+i where a and ,1J are positive
"
20. If the sequence {a } is defined by a"+1 =
constants, and n E N, with al and a any real numbers, thena+/3
a a 1 + (a -f- /3) a2
lima= =
a + 2/Q
21. Suppose that {an } and {bn } are two sequences such that {an } and an both converge.
b
Then bn or {b } must converge."
an
22. If a sequence of positive numbers is unbounded, then the sequence diverges to +X.
23. If {an } is a sequence and A is an accumulation point of the set S = {an I n E N}, then {an } must converge to A.
24. If S is a set of real numbers and A = sup S, then A --- c must be in S for any e>0.
an -F-1
25. If an 0 for any n E N, and {a} converges to 0, then limn_
an
<1.
26. The sequence {an } converges to zero if and only if {Ian j } converges to zero.
27. If line a = +aa , then limn- an = 1.
n- o0
2+aM 2
28. If {an } and {bn } are two sequences, where {an } converges to 0 and {b" } is bounded, then the sequence {a,, bn } converges to zero.
29. If {an } and {bn } are two sequences such that lim"_ a = limn bn, then there exists n* E N such that an = b" for all n ? n'.
30. If (a,1) is a sequence that converges to A, then the set S = {a" n E N) has an accumu- lation point.
31. If {an } is a sequence such that the set S = {an J n E N) is finite, then {an } converges.
32. If a = 1 4 7
III
(3n 2) then thesequence a converges to A where 0 A
1
233.
If A is an accumulation point of the set S, then A = sup S, provided that it exists.
34. The set S = {0, 1] has exactly two accumulation points.
35. If S is a set, s = sup S, and s S, then s must be an accumulation point of S.
at+a2+...+an
36. If {an } and {bn } are two sequences such that bn = for all n E N, then {an } converges to A if and only if [b} converges to A. (See summable sequencesn in Section 2.8.)
1irnn4
i J
4
5
233.
Sec. 2.8 * Projects 113 37. Any sequence has a unique limit.
38. If {an } and {bn } are two sequences such that {an } and {anbn } both converge, then {bn } converges.
39. If the sequence {a} is defined recursively by an+2 =n 1 + an+1,
for n E N with a1 = 1
an and a2 = 1, then {an } converges to the golden number.
40. There exists a sequence of rational numbers that has an irrational limit, and there exists a sequence of irrational numbers that has a rational limit.
41. If {an } diverges to +oo and {bn } is any sequence, then {an + bn } diverges to +oo.
42. If {an } and {bn } both diverge, then bn } diverges.
43. If {an } converges and {bn } diverges, then {an + b} diverges.
44. If lirnn an > 1, then an 1 eventually.
45. Every bounded sequence has a monotone subsequence.
46. If {an } and {bn } are sequences so that {an } and {anbn } both diverge, then {bn } diverges.
47. If {an } and {b } both diverge, then {anbn } diverges.
48. If all the convergent subsequences of {an } have the same limit, then {an } converges.
49. If {an } is convergent, then it is contractive.
50. The value of 2 + 2 + 2 is 2. (See Exercise 15 in Section 2.4.)
2.8* Projects