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).
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.
(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.
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.
(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.
Verify, using the definition of convergence of a sequence, that the following sequences converge to the proposed limit. (a) lim 2n+1
(c) lim sin()
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.
(a) Show that
√2, √2+ √2,√2+ √2+ √2,…
converges and find the limit.
(b) Does the sequence
converge? If so, find the limit.
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.
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.
Decide whether each of the following series converges or diverges:
(c) 1 – 1+砉-音+湯一西 +. 12
(d) 1+ + + + + +