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Sequence Convergence and Limit Proof

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Suppose {p_n} converges to p. Prove that there is at most one alpha for which the limit as n goes to infinity of
|p_n+1-p|/(|p_n-p|^alpha) is a positive finite number.

(See attachment for mathematical notation)

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The solution is half a page of Word with mathematical formulae written in Mathtype containing a step-by-step answer to the question.

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(Note: the statement is only true if we require the limit to be strictly positive. As an example, consider the sequence .
Then and , which is zero for any .) ...

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