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By Bloch S.J., et al. (eds.)

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Extra info for Applications of algebraic K-theory to algebraic geometry and number theory, Part 1

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5 are also known as the least limit 26 1 Prelude to Modem Analysis and the greatest limit, respectively, for S, and the following notations are used: £ = lim S, £ = lim S. With reference still to our proof o f the Bolzano-Weierstrass theorem, let £ = lim S. It is possible that no points of S are greater than £. Then £ is an upper bound for S. There is no value of <5 > 0 such that £ — <5 is also an upper bound for S since, £ being a cluster point for S , there must be (infinitely many) points o f S in (£ —<5, £ + <5).

They may be illustrated most simply using intervals. For example, let S be the open interval (0,1). The numbers —37, — 0 are lower bounds for S; the numbers 1, 7r, 72 are upper bounds. We have inf S = 0 , sup S = 1. Since inf S ^ S and sup£ ^ S, we see that min/S1 and max/S1 do not exist. If T is the closed interval [0,1], then i nfT = 0 E T, so m inT = 0; sup T = 1 E T, so m axT = 1. The interval (—co, 0) is bounded above but not below; its supremum is 0 . ). 2 Let S be a nonempty point set. A number £ is called a cluster point for S if every 8- neighbourhood of £ contains a point of S other than This definition does not imply that a cluster point for a set must be an element of that set.

From this it follows that a finite point set cannot have any cluster points. An infinite point set may or may not have cluster points. For exam­ ple, intervals have (infinitely many) cluster points, while the set Z of all integers has no cluster points. ) This leads us to the Bolzano-Weierstrass theorem, which provides a criterion for an infinite point set to have a cluster point. 3 (B o lza n o -W e ie rs tra ss T h e o r e m ) If S is a bounded infinite point set, then there exists at least one cluster point for S.

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