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By Sloughter D.

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7. Find a sequence In , n = 1, 2, 3, . , of closed intervals such that In+1 ⊂ In for n = 1, 2, 3, . . and ∞ In = ∅. 8. Find a sequence In , n = 1, 2, 3, . , of bounded, open intervals such that In+1 ⊂ In for n = 1, 2, 3, . . and ∞ In = ∅. 9. Suppose Ai ⊂ R, i = 1, 2, . . , n, and let B = that n i=1 Ai . Show n B= Ai . 10. Suppose Ai ⊂ R, i ∈ Z+ , and let ∞ B= Ai . i=1 Show that ∞ Ai ⊂ B. i=1 Find an example for which ∞ B= Ai . 11. Suppose U ⊂ R is a nonempty open set. For each x ∈ U , let Jx = where the union is taken over all (x − , x + δ), > 0 and δ > 0 such that (x − , x + δ) ⊂ U .

1. Suppose A and B are countable sets. Then the set C = A ∪ B is countable. Proof. Suppose A and B are disjoint, that is, A ∩ B = ∅. Let ϕ : Z+ → A and ψ : Z+ → B be one-to-one correspondences. Define τ : Z+ → C by τ (n) = ϕ ψ n+1 2 n 2 , , if n is odd, if n is even. 2) Then τ is a one-to-one correspondence, showing that C is countable. If A and B are not disjoint, then τ is onto but not one-to-one. However, in that case C has the cardinality of an infinite subset of Z+ , and so is countable. D.

Let nk be the smallest integer for which nk > nk−1 and xnk > t. Then {xnk }∞ k=1 is a subsequence of {xi }∞ which has a subsequential limit λ ≥ t. Since λ is also then a subsequeni=m tial limit of {xi }∞ , we have λ ∈ Λ and λ ≥ t > s, contradicting s = sup Λ. i=m Hence we must have lim sup xi = sup Λ. D. 27. Let Λ be the set of subsequential limits of a sequence {xi }∞ i=m . Show that lim inf xi = inf Λ. i→∞ 32 CHAPTER 2. 1. Given a sequence {ai }∞ i=m , let {sn }n=m be the sequence defined by n sn = ai .

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A primer of real analysis by Sloughter D.

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