Write 1. the definition of infimum by modeling it on the definition of supremum. State Lemma 1.3.7 for Infimums. 2. Let s = inf(A) and define -A = {-x : x A}. Prove that sup(-A) = -s. 3. Prove that inf {1 +
1 n2
: n N} = 1 using Lemma 1.3.7. 1+ 1+ 2, . . .) [Hint:
4. Find the limit of the sequence ( 2, 1 + 2, this sequence can be defined recursively.] 5. Find examples of the following, if possible:
(a) a sequence which is Cauchy, but not monotone. (b) a sequence which is monotone, but not Cauchy. (c) a sequence which is bounded, but not Cauchy. (d) two sequences (xn ) and (yn ) such that xn diverges, yn converges, and (xn + yn ) converges (e) two sequences (xn ) and (yn ) which both converge, but (xn yn ) diverges. (f) two sequences (xn and ) (yn ) which both diverge, but (xn yn ) converges. (g) A monotone sequence which diverges but has a convergent subsequence. (h) An unbounded sequence which has a Cauchy subsequence. 6. Prove that the following limits converge as indicated. (a) limn (b) (c)
1 =0 6n2 +1 3 3n+1 limn 2n+5 = 2 2 limn n+3
7. Suppose that lim xn = L and lim zn = L and for all n, xn yn zn . Prove that lim yn = L. [Note: you cannot use the order limit theorem, because we do not know if yn is convergent.] 8. Suppose that lim an = L and lim an = K. Prove that L = K. [Hint: Try a proof by contradiction or a proof by contrapositive.] 9. Suppose that (an ) and (bn ) are Cauchy sequences. Prove that cn = |an - bn | is also Cauchy using the definition of Cauchy.