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

ISBN-10: 0883853450

ISBN-13: 9780883853450

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

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It is not Cauchy since for any K > 0 you can find i, j ≥ K with d(yi , y j ) ≥ : pick i ≥ K so that yi = x is the start of a circuit and pick j > i so that y j is the midpoint of that circuit. This contradicts the fact that (X, d) is nonincremental, so the required δ must exist after all. At this stage we know that if (X, d) is a metric space then d uniformly equivalent to an ultrametric implies that d is nonincremental which in turn implies that d is equivalent to an ultrametric. It is natural to ask if either of these implications can be reversed.

If x, y ∈ X , say that x and y are -connected if there exist points x0 , x1 , . . , xn with x0 = x, xn = y, and d(xi , xi+1 ) < for i < n. Such a sequence of points is called an -chain connecting x and y. For x, y ∈ X let d ∗ (x, y) = inf{ | x and y are -connected}. Note that d ∗ (x, y) < implies that x and y are -connected (with respect to d). It is easy to check that d ∗ (x, y) ≥ 0, d ∗ (x, y) = d ∗ (y, x), and d ∗ (x, y) ≤ sup{d ∗ (x, z), d ∗ (z, y)} for all x, y, z ∈ X . This last inequality follows from the observation that the relation of being -connected is transitive.

Bn are complex numbers, then sin(z − b1 ) · · · sin(z − bn ) = cos (a1 + · · · + an − b1 − · · · − bn ) sin(z − a1 ) · · · sin(z − an ) n n j =1 k=1 1≤ j ≤n j =k + sin(ak − b j ) sin(ak − a j ) (A) cot(z − ak ) and cos(z − b1 ) · · · cos(z − bn ) = cos(a1 + · · · + an − b1 − · · · − bn ) cos(z − a1 ) · · · cos(z − an ) n n j =1 k=1 1≤ j ≤n j =k − 322 c sin(ak − b j ) sin(ak − a j ) (B) tan(z − ak ). THE MATHEMATICAL ASSOCIATION OF AMERICA [Monthly 117 (B) follows from (A) by replacing a j by a j + π2 and b j by b j + π2 for each j, 1 ≤ j ≤ n.