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 + for every > 0. (b) Two real numbers satisfy a < b if a < b + for every > 0.
(c) Two real numbers satisfy a b if and only if a < b 4- e for every e > 0.
Exercise 1.3.9. (a) If sup A < sup B, show that there exists an element b e 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 (0, l/n) = Ø. Notice that this demonstrates that the intervals in the Nested Interval Property must be closed for the conclusion 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 5n+4 ¯ 5 (b) lim n3+3 — o.— (c) lim
Exercise 2.3.1. Let 2 0 for all n e N.
(a) If (:cn) —+ 0, show that 0.
(b) If (Cn) —+ x, show that ( VG) —+ V.
Exercise 2.3.3 (Squeeze Theorem). Show that if n e N, and if limxn = limzn = l, then limyn = I as well.
run zn for all
Exercise 2.4.3. (a) Show that
converges and find the limit.
(b) Does the sequence
u, 2v6, 2 2v6, 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 e 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 e > 0, there exists an N such that if n N, then ISn+I —
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 (Cn) and (yn) are pseudo-Cauchy, then (Tn + yn) is pseudo-Cauchy as well.
Exercise 2.7.2. Decide whether each of the following series converges or diverges:
