21. Suppose (An) is a sequence of events. (a) If P (An) → 1 as n → ∞, prove there exists a subsequence {n} tend- ing to infinity such that P(nk Ank) > 0. (Hint: Use Borel-Cantelli.) (b) Show the following is false: Given > 0 such that P(An) ≥ , it follows that there exists a subsequence {n} tending to infinity such that P(nk Ank) > 0.
21. Suppose (An) is a sequence of events. (a) If P (An) → 1 as n → ∞, prove there exists a subsequence {n} tend- ing to infinity such that P(nk Ank) > 0. (Hint: Use Borel-Cantelli.) (b) Show the following is false: Given > 0 such that P(An) ≥ , it follows that there exists a subsequence {n} tending to infinity such that P(nk Ank) > 0.
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
Related questions
Question
![21. Suppose (An) is a sequence of events.
(a) If P (An) → 1 as n → ∞o, prove there exists a subsequence {n} tend-
ing to infinity such that P(nk Ank) > 0. (Hint: Use Borel-Cantelli.)
(b) Show the following is false: Given € > 0 such that P(An) ≥ €, it
follows that there exists a subsequence {n} tending to infinity such
that P (^k Ank) > 0.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F3a7e0308-dccb-4361-9acd-d47996ca0f5f%2F40f844d4-f0d0-4e4b-a212-4f4ec33235dc%2Fjsga0bb_processed.jpeg&w=3840&q=75)
Transcribed Image Text:21. Suppose (An) is a sequence of events.
(a) If P (An) → 1 as n → ∞o, prove there exists a subsequence {n} tend-
ing to infinity such that P(nk Ank) > 0. (Hint: Use Borel-Cantelli.)
(b) Show the following is false: Given € > 0 such that P(An) ≥ €, it
follows that there exists a subsequence {n} tending to infinity such
that P (^k Ank) > 0.
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