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I'm doing some exploratory afternoon reading, and I'm baffled by a minor detail in this paper. Here is the background:

A sequence $S = S_n$ is almost convergent to L if for any $\epsilon > 0$ we can find an integer n such that the average of n or more consecutive terms in the sequence is within $\epsilon$ of L. Formally, $\forall \epsilon > 0 \quad \exists N \ni \quad \displaystyle \left| \frac{1}{n} \sum_{i=0}^{n-1} S_{k-i} -L \right| < \epsilon \quad \forall n > N, \; k \in \mathbb{N}.$

I took the formal definition from here. Why didn't Professor Fikret Cunjalo say that we need $k \geq n$ or else we are looking at negative terms in the sequence? Is an interpretation of something like $S_{-3}$? Or maybe the author thinks that it would be obvious that $k \geq n$? (If so, it is not obvious to me!)


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The author might just be being sloppy. – Qiaochu Yuan Apr 6 '11 at 22:26
That's very possible ... I know they mostly write papers for other specialists who will not notice such errors, but being new to the subject little things can send me off on a tangent. – futurebird Apr 6 '11 at 22:34
The interpretation for $S_{-3}$ would be zero (or any other arbitrary choice). You can consider it an exercise to show that this doesn't matter for the sake of the definition. – Yuval Filmus Apr 9 '11 at 15:36

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