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6. Let E CR and E # 0. Prove if E is compact then every sequence in E has a subsequence that converges to a point in E.

6. Let E CR and E # 0. Prove if E is compact then every sequence in E has a subsequence that converges to a point in E.

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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I need help with #6. Thank you.

1. Give an example of a sequence of real numbers with the following properties. Proof ) We h (a) Cauchy but not monotone (b) Monotone but not Cauchy (c) Bounded but not Cauchy nce. 2. Use the definitions of Cauchy and bounded sequences to prove that all Cauchy se- quences are bounded. 3. For each of the following sequences find the set S of subsequential limit points, the limit superior and limit inferior. Include too as possible limits. (a) (In) = (0, 1, 2, 0, 1, 3, 0, 1, 4, ...) (b) xn = n(2+ (-1)"). (c) xn = n · coS 2 4. A sequence (xn) is said to oscillate if lim inf xn < lim sup rn. Prove or disprove the following. (a) Every oscillating sequence diverges. (b) Every divergent sequence oscillates. (c) Every oscillating sequence has a convergent subsequence. 5. Prove or disprove: If (x„) and (Yn) are subsequences of each other, then x, = Yn- 6. Let E C R and E + 0. Prove if E is compact then every sequence in E has a subsequence that converges to a point in E.
Transcribed Image Text:1. Give an example of a sequence of real numbers with the following properties. Proof ) We h (a) Cauchy but not monotone (b) Monotone but not Cauchy (c) Bounded but not Cauchy nce. 2. Use the definitions of Cauchy and bounded sequences to prove that all Cauchy se- quences are bounded. 3. For each of the following sequences find the set S of subsequential limit points, the limit superior and limit inferior. Include too as possible limits. (a) (In) = (0, 1, 2, 0, 1, 3, 0, 1, 4, ...) (b) xn = n(2+ (-1)"). (c) xn = n · coS 2 4. A sequence (xn) is said to oscillate if lim inf xn < lim sup rn. Prove or disprove the following. (a) Every oscillating sequence diverges. (b) Every divergent sequence oscillates. (c) Every oscillating sequence has a convergent subsequence. 5. Prove or disprove: If (x„) and (Yn) are subsequences of each other, then x, = Yn- 6. Let E C R and E + 0. Prove if E is compact then every sequence in E has a subsequence that converges to a point in E.
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