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By Daniel J. Velleman

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

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But d ∗ is an ultrametric, so repeated application of the ultrametric inequality yields d ∗ (x, y) < γ . Thus d(x, y) < . We can thus infer that if d is uniformly equivalent to an ultrametric then ∀ > 0 ∃ δ > 0 (x, y δ-connected ⇒ d(x, y) < ). (∗) Now let X = {(n, m/n) | m, n integers with n ≥ 1 and 0 ≤ m ≤ n} (see Figure 1). ✻ 1 t t t t t t t t t 1 2 3 q q q q q q ✲ Figure 1. Consider X as a subset of the Cartesian plane and let d be the induced Euclidean metric. X consists of infinitely many columns where each column is distance 1 from its neighboring columns.

Wn ), k=0 where em (w1 , . . 7) for some constants Dk . 7) by u − wk and then letting u → wk , which gives Dk = 2 wk Wn,k . 7) is n j =1 u + wj =1+ u − wj April 2010] n Wn,k k=1 2 wk =1+ u − wk n Wn,k k=1 u + wk − u − wk ` LA HERMITE TRIGONOMETRIC IDENTITIES A n Wn,k . 8), it reduces to n (−1)n = 1 − 2 Wn,k k=1 so that n Wn,k = k=1 nπ 1 − (−1)n = sin2 . 8) we have Lemma 2, which is Hermite’s theorem in disguise. To see this, replace u by e2i z and each wk by e2iak , and observe that e2i z + e2iak e2i z + e2iak e−i(z+ak ) ei(z−ak ) + e−i(z−ak ) = × = = −i cot(z − ak ).

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. The analogous property fails for quasi-Cauchy sequences. In fact, it is precisely the Cauchy sequences which have the property that all of their subsequences are quasi-Cauchy.

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