Number Theory: An Introduction to Mathematics

26 I The Expanding Universe of Numbers Proposition 25Any fundamental sequence of real numbers is convergent. Proof If{an}is a fu ...

4 Metric Spaces 27 4 Metric Spaces The notion of convergence is meaningful not only for points on a line, but also for points in ...

28 I The Expanding Universe of Numbers In any metric space there is a naturaltopology. A subsetGof a metric spaceE isopenif for ...

4 Metric Spaces 29 (iii) LetE=C(I)be the set of all continuous functionsf:I→R,where I=[a,b]={x∈R:a≤x≤b} is an interval ofR, and ...

30 I The Expanding Universe of Numbers Here the triangle inequality holds in the stronger form d(a,b)≤max[d(a,c),d(c,b)]. This m ...

4 Metric Spaces 31 By generalizing the M ́eray–Cantor method of extending the rational numbers to the real numbers, Hausdorff (1 ...

32 I The Expanding Universe of Numbers In example (ii), the completeness ofFn 2 is trivial, since any fundamental sequence is ul ...

4 Metric Spaces 33 Consequently, ifn>m≥0, d(xn,xm)≤d(xn,xn− 1 )+d(xn− 1 ,xn− 2 )+···+d(xm+ 1 ,xm) ≤(θn−^1 +θn−^2 +···+θm)d(x ...

34 I The Expanding Universe of Numbers We will also use the notion of norm of a linear map. IfA:Rn→Rmis a linear map, itsnorm|A| ...

4 Metric Spaces 35 Ifx 1 ,x 2 ∈U,then |fy(x 2 )−fy(x 1 )|= ∣ ∣ ∣ ∣ ∫ 1 0 f′(( 1 −t)x 1 +tx 2 )(x 2 −x 1 )dt ∣ ∣ ∣ ∣ ≤|x 2 −x 1 | ...

36 I The Expanding Universe of Numbers Hence |ψ(y)−ψ(η)−G(y−η)|/|y−η|≤ 2 |A−^1 ||G||φ(x)−φ(ξ)−F(x−ξ)|/|x−ξ|. If|y−η|→0, then|x−ξ ...

4 Metric Spaces 37 Proof Ifx(t) is a solution of the differential equation (1) which satisfies the initial condition (2), then b ...

38 I The Expanding Universe of Numbers Proposition 28 only guarantees the local existence of solutions, but this is in the natur ...

5 Complex Numbers 39 Of courseE(t)=etis theexponential function. We will now adopt the usual notation, but we remark that the de ...

40 I The Expanding Universe of Numbers whereaandbare real numbers. The set of all complex numbers is customarily denoted byC. We ...

5 Complex Numbers 41 andIzrespectively. Complex numbers of the formiy,wherey∈R, are said to be pure imaginary. It is worth notin ...

42 I The Expanding Universe of Numbers |z+w|^2 ≤|z|^2 + 2 |z||w|+|w|^2 =(|z|+|w|)^2 , and (iii) follows by taking square roots. ...

5 Complex Numbers 43 f(z)=a 0 zn+a 1 zn−^1 +···+an, wherea 0 ,a 1 ,...,an∈C,n≥1anda 0 =0, has a complex root. Thus by adjoining ...

44 I The Expanding Universe of Numbers discB={z∈C:|z−ζ|≤δ}is contained inGand contains no point ofEexceptζ.If S={z∈C:|z−ζ|=δ}is ...

5 Complex Numbers 45 uses the result, due to Kronecker (1887), that a polynomial with coefficients from an arbitrary fieldKdecom ...