## Description

Exercises 1.2.10, 1.3.9, 1.4.3, 1.5.6, 2.2.2 (b), 2.3.1 (b), 2.3.3, 2.4.3 (b),

2.5.5, 2.6.5, 2.7.2 (c), (e).

## Exercise 1.2.10.

Decide which of the following are true statements. Provide a short justification for those that are valid and a counterexample for those that are not:

(a) Two real numbers satisfy a <b if and only if a <b+e for every € > 0. (b) Two real numbers satisfy a <b if a < be for every € > 0.

(c) Two real numbers satisfy a <b if and only if a <b+€ for every € > 0.

## Exercise 1.3.9.

(a) If sup A < sup B, show that there exists an element be B that is an upper bound for A.

(b) Give an example to show that this is not always the case if we only assume sup A≤ sup B.

## Exercise 1.4.3.

Prove that 1(0,1/n) = 0. Notice that this demonstrates that the intervals in the Nested Interval Property must be closed for the con- clusion of the theorem to hold.

## Exercise 1.5.6.

(a) Give an example of a countable collection of disjoint open intervals.

(b) Give an example of an uncountable collection of disjoint open intervals, or argue that no such collection exists.

## Exercise 2.2.2.

Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. (a) lim 2n+1

5n+4

(b) lim

2n2 n3+3

= 0.

(c) lim sin()

=

0.

## Exercise 2.3.1.

Let n > 0 for all n e N.

(a) If (n) 0, show that (√) → 0.

(b) If (n), show that (√) →√x.

## Exercise 2.3.3 (Squeeze Theorem).

Show that if nyn zn for all nЄN, and if liman lim z = 1, then lim yn as well.

## Exercise 2.4.3.

(a) Show that

√2, √2+ √2,√2+ √2+ √2,…

converges and find the limit.

(b) Does the sequence

V

√2,√2√2,√/2/2√2….

converge? If so, find the limit.

## Exercise 2.5.5.

Assume (an) is a bounded sequence with the property that every convergent subsequence of (an) converges to the same limit a € R. Show that (an) must converge to a.

## Exercise 2.6.5.

Consider the following (invented) definition: A sequence (sn) is pseudo-Cauchy if, for all > 0, there exists an N such that if n > N, then Sn+1

Sn❘ < €.

Decide which one of the following two propositions is actually true. Supply a proof for the valid statement and a counterexample for the other.

(i) Pseudo-Cauchy sequences are bounded.

(ii) If (2) and (v) are pseudo-Cauchy, then (a,+ n) is pseudo-Cauchy as well.

## Exercise 2.7.2.

Decide whether each of the following series converges or diverges:

(2) ΣΕΞ1

n=1 2″+n

(1) ΣΕΞ1

sin(n) n2

7

(c) 1 – 1+砉-音+湯一西 +. 12

(d) 1+ + + + + +

(e) 1-+-+-+-+..

3