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Unformatted text preview: Modern Analysis 1
Midterm
Answer at most ﬁve questions.
1. Let S be a complete ordered set. For each λ ∈ Λ let Aλ ⊂ S be nonempty
and let A = ∪λ∈Λ Aλ be bounded above; let α = sup A and for each λ ∈ Λ
let αλ = sup Aλ . Prove or disprove:
α = sup{αλ : λ ∈ Λ}.
2. (i) State the CauchySchwarz inequality, with a precise necessary and
suﬃcient condition for equality.
N
2
(ii) Find the minimum value of
n=1 an given that the N real numbers
a1 , . . . , aN have sum equal to 1.
3. Let
K = {1/n : Z n > 0} ∪ {0} with the usual (Euclidean) metric.
(i) Prove directly that K is compact.
(ii) Which subsets of K are compact? Prove.
(iii) Which subsets of K are precompact? Explain.
4. Let A and B be disjoint, nonempty compact subsets of a metric space.
Deﬁne
δ = inf {d(a, b) : a ∈ A, b ∈ B }.
Prove that there exist a0 ∈ A and b0 ∈ B such that δ = d(a0 , b0 ).
5. (i) Prove that a real sequence that is increasing and bounded above converges.
√
(ii) Prove that the real sequence (an )∞ deﬁned by a0 = 2 and
n=0
√
n 0 ⇒ an+1 = 2 + an
converges and ﬁnd its limit.
6. Let (an )n∈N be a sequence of positive reals with the property that an → 0
as n → ∞. Prove that there exists a permutation π of N such that the
rearranged sequence (aπ(n) )n∈N is decreasing:
aπ(0) aπ(1) .... Decide (with justiﬁcation) what happens to aπ(n) as n → ∞.
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This note was uploaded on 11/09/2011 for the course MAA 5228 taught by Professor Robinson during the Fall '09 term at University of Florida.
 Fall '09
 Robinson

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