Download A Treatise On The Differential Calculus with numerous by Isaac Todhunter PDF

By Isaac Todhunter

This Elibron Classics ebook is a facsimile reprint of a 1864 version through Macmillan and Co., Cambridge and London.

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Extra resources for A Treatise On The Differential Calculus with numerous examples

Example text

AN |}. If j is any natural number, then either 1 ≤ j ≤ N , in which case |aj | ≤ L, or else j > N , in which case |aj | ≤ K. Set M = max{K,L}. Then, for any j, |aj | ≤ M as required. In what follows we shall use an interesting and not entirely obvious version of the triangle inequality. You know the triangle inequality as |a + b| ≤ |a| + |b| . But let us instead write |a| = |(a + b) − b| = |(a + b) + (−b)| ≤ |a + b| + | − b| = |a + b| + |b| . From this we conclude that |a + b| ≥ |a| − |b| .

If x is a rational upper bound for C and c ∈ C then −x ∈ −C so −C is not empty. By its very definition, C + (−C) ⊆ 0. Further, if z ∈ 0 and c ∈ C we set c′ = z − c. Then c′ ∈ −C and z = c + c′ . Hence 0 ⊆ C + (−C). We conclude that C + (−C) = 0. Having verified the axioms for addition, we turn now to multiplication. 17 If C and D are cuts then we define the product C ·D as follows: • If C, D > 0 then C · D = {q ∈ Q : q < c · d for some c ∈ C, d ∈ D with c > 0, d > 0 } • If C > 0, D < 0 then C · D = − (C · (−D)) • If C < 0, D > 0 then C · D = − ((−C) · D) • If C, D < 0 then C · D = (−C) · (−D) • If either C = 0 or D = 0 then C · D = 0.

40 Consider the sequence {aj } in the last example. Let ǫ > 0. Choose j even so that j > (1 − ǫ)/ǫ. Then j2 > 1−ǫ. j2 + j Now again choose ǫ > 0. Choose j odd so that j > (1 − ǫ)/ǫ. Then − j2 < −1 + ǫ . +j j2 36 CHAPTER 2. SEQUENCES Exercises 1. Consider {aj } both as a sequence and as a set. How are the lim sup and the sup related? How are the lim inf and the inf related? Give examples. 2. Let {aj } be a sequence of positive numbers. How are the lim sup and lim inf of {aj } related to the lim sup and lim inf of {1/aj }?

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