Download A treatise on the analytic geometry of three dimensions by George Salmon PDF

By George Salmon

Quantity: 2 writer: London Longmans, eco-friendly topics: Geometry, Analytic -- sturdy Surfaces Notes: this can be an OCR reprint. there's typos or lacking textual content. There aren't any illustrations or indexes. if you purchase the overall Books variation of this ebook you get loose trial entry to Million-Books.com the place you could choose from greater than 1000000 books at no cost. it's also possible to preview the ebook there.

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Extra resources for A treatise on the analytic geometry of three dimensions

Example text

Then {Aj } is an increasing sequence (since, as j becomes large, we are taking the infimum of a smaller set of numbers), so it has a limit (either a finite limit or ±∞). We define the limit infimum of {aj } to be lim inf aj = lim Aj . j→∞ It is common to refer to this number as the lim inf of the sequence. Likewise, let Bj = sup{aj , aj+1 , aj+2 , . . } . Then {Bj } is a decreasing sequence (since, as j becomes large, we are taking the supremum of a smaller set of numbers), so it has a limit (either a finite limit of ±∞).

20. 8. 6. 9. Give an example of a decreasing sequence that converges to π. 28 CHAPTER 2. SEQUENCES 10. Prove the following result, which we have used without comment in the text: Let S be a set of real numbers which is bounded above and let t = sup S. For any ǫ > 0 there is an element s ∈ S such that t − ǫ < s ≤ t. ) Formulate and prove a similar result for the infimum. 11. Let {aj } be a sequence of real or complex numbers. Suppose that every subsequence has itself a subsequence which converges to a given number α.

Notice that the lim sup or lim inf of a sequence can be ±∞. 34 What is the intuitive content of this definition? For each j, Aj picks out the greatest lower bound of the sequence in the j th position or later. So the sequence {Aj } should tend to the smallest possible limit of any subsequence of {aj }. Likewise, for each j, Bj picks out the least upper bound of the sequence in the j th position or later. So the sequence {Bj } should tend to the greatest possible limit of any subsequence of {aj }.

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