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Cantor's Theorem Let {F} be a sequence of non-empty subsets of satisfying:(i) each F is closed, and 12 (ii) F₁ = F₂ =.... 2 00 (a) If one of the sets F is bounded then n F#Ø 72 n=1 (b) If the sequence of the sets {F} also satisfies (iii) diamF → 0, then ʼn F is a 12 12 n=1 single point. Consider the three restrictions (i), (ii), and (iii) placed on the sets {F} in Cantor's Theorem. (a) Find a sequence of sets {F} that satisfies (i) and (ii), but n F_=. 00 72 n=1 (b) Find a sequence of sets {F} that satisfies (i) and (iii), but n F =0. 22 n=1 (c) Find a sequence of sets {F} that satisfies (ii) and (iii), but ʼn F₁ = 22 n=1

Cantor's Theorem Let {F} be a sequence of non-empty subsets of satisfying:(i) each F is closed, and 12 (ii) F₁ = F₂ =.... 2 00 (a) If one of the sets F is bounded then n F#Ø 72 n=1 (b) If the sequence of the sets {F} also satisfies (iii) diamF → 0, then ʼn F is a 12 12 n=1 single point. Consider the three restrictions (i), (ii), and (iii) placed on the sets {F} in Cantor's Theorem. (a) Find a sequence of sets {F} that satisfies (i) and (ii), but n F_=. 00 72 n=1 (b) Find a sequence of sets {F} that satisfies (i) and (iii), but n F =0. 22 n=1 (c) Find a sequence of sets {F} that satisfies (ii) and (iii), but ʼn F₁ = 22 n=1

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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Cantor's Theorem Let {F} be a sequence of non-empty subsets of satisfying:(i) each F is closed, and 12 (ii) F₁ = F₂ =.... 2 00 (a) If one of the sets F is bounded then n F#Ø 72 n=1 (b) If the sequence of the sets {F} also satisfies (iii) diamF → 0, then ʼn F is a 12 12 n=1 single point. Consider the three restrictions (i), (ii), and (iii) placed on the sets {F} in Cantor's Theorem. (a) Find a sequence of sets {F} that satisfies (i) and (ii), but n F_=. 00 72 n=1 (b) Find a sequence of sets {F} that satisfies (i) and (iii), but n F =0. 22 n=1 (c) Find a sequence of sets {F} that satisfies (ii) and (iii), but ʼn F₁ =Ø 22 n=1
Transcribed Image Text:Cantor's Theorem Let {F} be a sequence of non-empty subsets of satisfying:(i) each F is closed, and 12 (ii) F₁ = F₂ =.... 2 00 (a) If one of the sets F is bounded then n F#Ø 72 n=1 (b) If the sequence of the sets {F} also satisfies (iii) diamF → 0, then ʼn F is a 12 12 n=1 single point. Consider the three restrictions (i), (ii), and (iii) placed on the sets {F} in Cantor's Theorem. (a) Find a sequence of sets {F} that satisfies (i) and (ii), but n F_=. 00 72 n=1 (b) Find a sequence of sets {F} that satisfies (i) and (iii), but n F =0. 22 n=1 (c) Find a sequence of sets {F} that satisfies (ii) and (iii), but ʼn F₁ =Ø 22 n=1
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