6.2 Determine the Limits of the Following Sequences and Then Prove Your Claim

Homework 6

6.1 Prove the following:

(a) (b)

6.2 Determine the limits of the following sequences and then prove your claim:

(a) (b)

6.3 Let (sn ) be a sequence of nonnegative real numbers and suppose that

. Prove that .

6.4 (a)Consider three sequences (an), (bn) and (sn) such that for all n

and . Prove that .

(b) Suppose that (sn) and (tn) are sequences such that for all n and

. Prove that .

6.5  Let (sn) be a sequence in R.

(a)  Prove that if and only if .

(b)  Observe that if , then does not exists but exist.

6.6 Let (sn) be a convergent sequence and suppose that . Prove that

there exists a number N such that n > N implies sn > a.

6.7 Give an example of two divergent sequences (xn) and (yn) such that:

(a) Their sum (xn+ yn) converges

(b) Their product (xn yn) converges

6.8 Show that if (xn) and (yn) are sequences such that (xn) and (xn+ yn) are

convergent, then (yn) is convergent.

6.9 Show that if (xn) and (yn) are sequences such that (xn) converges to

and (xn yn) converges, then (yn) converges.

6.10 Solve example 3.7(ii)

Homework 7

7.1  Let s1 = 1 and for let .

(a)  List the first four terms of (sn).

(b)  It turns about that (sn) converges. Assume this fact and prove that the limit is

7.2  Suppose there exists No such that for all n > No

(a)  Prove that if , then .

(b)  Prove that if , then .

7.3  (a) Show that if and k > 0, then .

(b) Show that if and only if .

(c)  Show that if and k < 0, then

7.4  Show that if and if (tn) is a bounded sequence, then

.

7.5  Prove theorem 3.17

Homework 8

8.1  Let S be a bounded nonempty subset of R and suppose . Prove that

there is a nondecreasing sequence (sn) of points in S such that .

8.2  Let s1 = 1 and for .

(a)  find s2, s3 and s4.

(b)  Use mathematical induction to show that for all n.

(c)  Show that (sn) is a nonincreasing sequence.

(d)  Show that exists and find .

8.3  Give an example of a bounded sequence that is not a Cauchy sequence.

8.4  Show directly from the definition that the sequence is a Cauchy

sequence.

8.5  Show directly from the definition that if (xn) and (yn) are Cauchy sequences, then (xn + yn) and (xn yn) are Cauchy sequences.

8.6  Give an example of an unbounded sequence that has a convergent

subsequence.

8.7  Show that the sequence is divergent.

8.8 Prove theorem 3.26(ii).

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