Download American Mathematical Monthly, volume 117, number 4, April by Daniel J. Velleman PDF

By Daniel J. Velleman

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Extra info for American Mathematical Monthly, volume 117, number 4, April 2010

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He has now published articles on calculus and on trigonometry in this M ONTHLY, and he is wondering whether he will be able to sell them on addition of fractions. edu April 2010] ` LA HERMITE TRIGONOMETRIC IDENTITIES A 327 Quasi-Cauchy Sequences David Burton and John Coleman Beginning students often misunderstand the definition of Cauchy sequences when they first encounter it in an introductory real analysis course. In particular, many students fail to understand that it involves far more than that the distance between successive terms is tending to zero.

It follows that (The first inequality is a consequence of (26) and (29) since the sets {a, b, c} and {x, y, z} are disjoint. ) We now complete our proof, much as in Theorem 1, by observing that the inequality b y g(t) dt < t z c g(t) dt + t x a g(t) dt t (35) is valid for all strictly convex functions g : [c, x] → R, courtesy of (24) and (25). Remark 3. The hypotheses of Theorem 3, unlike those of Theorem 1, are not translation invariant. In particular, it is impossible to derive (22) from (24)–(26) unless the parameters a, b, c, x, y, z are all positive.

Xn is Cauchy if given any > 0 there exists an integer K > 0 such that m, n ≥ K implies |xm − xn | < . 2. xn is quasi-Cauchy if given any > 0 there exists an integer K > 0 such that n ≥ K implies |xn+1 − xn | < . Trivially, Cauchy sequences are quasi-Cauchy. The converse is easily seen to be false. The most quotable (though not the simplest) counterexample is provided by the sequence of partial sums of the harmonic series. Cauchy sequences have the property that any subsequence of a Cauchy sequence is Cauchy.

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