Indian Institute of Technology Guwahati

Department of Mathematics

MA 101 Mathematics-I Tutorial Sheet-2

Calculus Date: 05-Oct-2012

Topics Covered: Monotonic sequences, Subsequences and Bolzano-Weierstrass theorem, Cauchy

sequences.

1. For each of the following sets, check whether it is bounded below, bounded above. Also nd

supremum and inmum if they exist. Do they have maximum and minimum?

(a) S ={

1m 1

n| m,n N}(b) S =

{1 1

n| n N}(c) S = {sin(n) | n N}(d) S = {x R | (x a)(x b)(x c) < 0}with a < b < c.

2. Show that the supremum (and inmum) of a subset S R is unique, if it exists.3. Let A be a nonempty subset of R and be a real number. If = supA then show that a for all a A and for any > 0 there is some a0 A such that < a0.What happens if is a maximum?

4. Let a, b R with a > 0. Show that there exists n N such that na > b. (This is equivalentto the Archimedean property.)

5. For each of the following sequences determine whether it is monotonic. Also check whether it

is Cauchy.

(a)

(n1n+1

)(b) Fibonacci sequence(c) an+1 =

2an 1 for various values of a1.

6. Let (xn) be a monotonically increasing sequence such that xn 1 for all n N. Can you ndsupremum of the set {xn | n N}?7. Let (xn) be a sequence of nonzero real numbers such that it does not have any convergent

subsequence. Show that the sequence

(1xn

)is convergent. What is the limit?

8. Show that every sequence has a monotonic subsequence.

9. Complete the proof of Cauchy Convergence Criterion.

10. (Nested Interval Theorem) For each n N, let In = [an, bn] be a non-empty boundedinterval of R such that

I1 I2 In In+1 and lim

n(an bn) = 0. Show that

nN

In contains exactly one point.