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end3r7
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Homework Statement
I need to prove if these are true, and provide a counter if they are false. Please tell me if I have these right, I think they are all true.
(i) Let s(n) be a sequence s.t. [tex]\lim_{n\rightarrow\infty} (s(n+1) - s(n)) = 0[/tex]. Then s(n) must converge.
(ii) Let s(n) be a sequence s.t. [tex]|s(n+1) - s(n)| < \frac{1}{n}[/tex] for all n. Then s(n) must converge.
(iii) If [a(n)]^2 --> A^2, then either a(n) --> A or a(n) --> -A
(iv) If [a(n)]^3 --> A^3, then a(n) --> A
(v) If [s(n)]^2 --> S^2 and [tex]\lim_{n\rightarrow\infty} (s(n+1) - s(n)) = 0[/tex], then either s(n) --> S or s(n) --> -S
Homework Equations
The Attempt at a Solution
(i) TRUE. It's a cauchy sequence. Proof involves showing that limit sup <= limit inf which implies limit sup = limit inf which in turn implies a limit exists.
(ii) TRUE. A cauchy sequence with epsilon = 1/n
(iii) TRUE. Trivial using the limit of the product of two convergent sequences is the product of the two limits.
(iv) TRUE. Same way.
(v) ? Just look back at (i) and (iii), I guess...
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