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By Daniel J. Velleman

ISBN-10: 0883853450

ISBN-13: 9780883853450

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3) as G(z) = F(1, cot z) (sin z)n F(1, cot z) = , cos a j − cos z sin a j ) A(1, cot z) n j =1 (sin z where A(u, v) is defined in Lemma 3. 1) and Lemma 3, this is bounded as the imaginary part of z becomes infinite, and it is also bounded in the real direction by periodicity. As in the proof of Theorem 3, then, we must have F(sin z, cos z) = sin(z − a1 ) · · · sin(z − an ) n k=1 F(sin ak , cos ak ) cot(z − ak ) + C, 1≤ j ≤n sin(ak − a j ) j =k because the difference between the left and right sides is a bounded entire function.

Prove that a quasi-Cauchy sequence of real numbers is Cauchy if and only if it has exactly one cluster point. 5. Prove that the set of cluster points of a quasi-Cauchy sequence in R is closed and connected. 6. Prove that any closed, connected set in R is the cluster point set of some quasiCauchy sequence. 7. Call a set A in a metric space (X, d) pseudoconnected if it has the property that any two points in A are -connected in (A, d ) for all > 0 (where the d in (A, d ) is the restriction of d to A).

Glaisher was editing The Messenger of Mathematics and The Quarterly Journal of Pure and Applied Mathematics by this time. 1), which is surprising at first sight, as it would have been a good fit in the Messenger— he often published trigonometric identities there in this period, for example [9] and [10]—but it is not hard to guess the reason why. In 1876 he published a note [3] by Cayley there that makes [19] look like Anna Karenina. ) Expanding on the first column, it is easy to see that this determinant is sin(A + F) sin(B + F) sin(C + F) sin(H − G) + sin(A + G) sin(B + G) sin(C + G) sin(F − H ) + sin(A + H ) sin(B + H ) sin(C + H ) sin(G − F).

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American Mathematical Monthly, volume 117, number 4, April 2010 by Daniel J. Velleman

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