Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN: 9780470458365
Author: Erwin Kreyszig
Publisher: Wiley, John & Sons, Incorporated
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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.
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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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