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
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.