By V. P. Havin, S.V. Hruscev, N.K. Nikol'skii

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Then there exists at least a point x common to all intervals, x E Proof. , the sequence {an} is increasing. n~=l Cn. 2. , the sequence {bn } is decreasing. (iii) an ~ bm for all n and m. 22 they have finite limits, an if, bn ! L, and we have Vn,m In particular E N. nen. 32,. 31 we have [l, L] = n;:"=l en. h. Cauchy criterion Except for monotone sequences we cannot state that a sequence converges without involving its limit in advance. 33 Definition. :3 n such that IXh - xkl < f. Vh,k ~ n. 7) Notice that dk can be understood as the length of the interval spanned by all the elements of the sequence {xn} but the first k.

In fact for n = 0 we have (1 + h)O = 1 = 1 + O· h. If now n E iii and we assume (1 + h)n ~ 1 + nh, then (1 + h)n+1 + h)(1 + h)n ~ (1 + h)(1 + nh) 1 + (n + l)h + h 2 ~ 1 + (n + l)h. 35 Example (Arithmetic and quadratic mean). Let us show by induction that ( 1 2 n - Laj n j=l ) 1 n ~ - Lar n j=1 For n = 1 the claim is trivial. Suppose the claim true for n and let us prove it for n + 1. 2) and the inductive assumption then yield n n La~ j=1 + (n+ l)a~+1 n+l = (n+ 1) L a~. 36 Example (Sum of the first n naturals).

Giuseppe Peano (18581932) and the frontispiece of A rithmetices Principia, Torino, 1889. "'T_ _ ........... ~ .. , ' ... "~"' .. e. , if the successors of a and b are equal, then so are a and b, (v) if A c N is such that 0 E A, and a E A implies O'(a) E A, then A = N. Axioms (i)-(v) were introduced by Giuseppe Peano (1858-1932), who also showed how one can derive from them the entire arithmetic: they are known as Peano's axiom. Starting from natural numbers one can build successively the system of signed integers, denoted by Z, of rationals Q and of reals JR.