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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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