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If ∞ then bi converges, i=k ai converges. i=m ∞ Proof. 3 We need only show that ∞ the nth partial sum of ai converges. Let sn be i=N ∞ ai and let tn be the nth partial sum of i=N bi . 8) for every n ≥ N , so {sn }∞ n=N is a nondecreasing sequence. 9) i=N for every n ≥ N . Thus {sn }∞ n=N is a nondecreasing, bounded sequence, and so converges. D. 4. Suppose i=m i=k ∞ exists an integer N such that 0 ≤ ai ≤ bi whenever i ≥ N . If ∞ then ai diverges, i=k bi diverges. i=m ∞ Proof. 3 we need only show that ∞ the nth partial sum of bi diverges.

We let √ x denote the number s in the previous exercise, the square root of x. 5. Suppose {xi }i∈I is a convergent sequence in R, α is a real number, and L = lim xi . Then the sequence {αxi }i∈I converges and i→∞ lim αxi = αL. 33) i→∞ Proof. If α = 0, then {αxi }i∈I clearly converges to 0. So assume α = 0. 34) |α| whenever i > N . Then for any i > N we have |αxi − αL| = |α||xi − L| < |α| Thus lim αxi = αL. i→∞ |α| = . D. 1. 6. Suppose {xi }i∈I and {yi }i∈I are convergent sequences in R with L = lim xi and M = lim yi .

Since [a, b] is compact, there exists a finite subcover of this cover. This subcover is either of the form {Uβ : β ∈ B} or {Uβ : β ∈ B} ∪ {V } for some B ⊂ A. 12) β∈B in the latter case, we have K ⊂ [a, b] \ V ⊂ Uβ . 13) β∈B In either case, we have found a finite subcover of {Uα : α ∈ A}. D. 56 CHAPTER 4. 4. Show that if K is compact and C ⊂ K is closed, then C is compact. 3. If K ⊂ R is compact, then K is closed. Proof. Suppose x is a limit point of K and x ∈ / K. For n = 1, 2, 3, . , let Un = Then −∞, x − 1 n ∪ x+ 1 , +∞ .

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