D efinition 18 A sequence z n converges to w C if for every \u03b5 0 there exists a

D efinition 18 a sequence z n converges to w c if for

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D efinition 1.8. A sequence { z n } converges to w C if for every ε > 0 there exists a number N = N ε > 0 such that | z n - w | < ε as soon as n > N . Notation: lim n →∞ z n = w or z n w , n → ∞ . A sequence { z n } is said to be a Cauchy sequence (or Cauchy), if for every ε > 0 there is a number N = N ε > 0 such that | z n - z m | < ε as soon as n , m > N . In other words, | z n - z m | → 0 as n , m → ∞ .
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6 1. GEOMETRY, TOPOLOGY AND ANALYSIS IN THE COMPLEX PLANE This definition is exactly the same as in Real Analysis except that the sequence and the limit are now complex-valued. P roposition 1.9. z n w i ff Re z n Re w and Im z n Im w as n → ∞ . P roof . Write: | Re z n - Re w | 2 + | Im z n - Im w | 2 = | z n - w | 2 . The right-hand side tends to zero as n → ∞ , and the left-hand side is non-negative. Therefore, the left-hand side and the right-hand side tend to zero at the same time, as required. C orollary 1.10. If z n w, then z n w and | z n | → | w | . P roposition 1.11. A sequence { z n } converges i ff it is Cauchy. P roof . The real-valued sequences Re z n , Im z n converge i ff they are Cauchy, see Analysis 1. In view of this proposition we say that the set of complex numbers C is complete. 2.2. Sets of the complex plane. D efinition 1.12. Let z 0 be a complex number and r > 0 be a real number. Then the set γ ( z 0 , r ) = { z : | z - z 0 | = r } = { z = z 0 + re i φ , φ ( - π, π ] } , is called circle centered at z 0 C of radius r > 0. The set D ( z 0 , r ) = { z C | | z - z 0 | < r } is called an r - neighbourhood of z 0 or an open disk of radius r centered at z 0 . The set D ( z 0 , r ) = { z C | | z - z 0 | ≤ r } is called the closed disk of radius r centered at z 0 . The set D 0 ( z 0 , r ) = { z C : 0 < | z - z 0 | < r } is called a punctured r -neighbourhood. The set Π ± = { z : ± Im z > 0 } is called the upper (lower) half-plane . Let S C be a set of complex numbers. D efinition 1.13. A point z S is said to be an interior point of the set S if there is a r > 0 such that D ( z , r ) S . The notation int S is used for the set of all interior points of S . The set S is said to be open if S = int S , that is S consist of interior points, that is for each z S there is a number r > 0(possibly depending on z ) such that D ( z , r ) S .
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3. FUNCTIONS, THEIR LIMITS AND CONTINUITY 7 E xample . (1) The set Π + is open. Indeed, let z = x + iy Π + , y > 0. For any w D ( z , y ) we have Im w = y + Im( w - x ) y - | w - x | > 0 , and hence w Π + . (2) The open disk D ( a , r ) is open. Indeed, let z D ( a , r ). Denote ε = r - | z - a | > 0. Then for any w D ( z , ε ) we have | w - a | ≤ | w - z | + | z - a | < ε + | z - a | = r , as required. (3) Let a , b C . The set Ω = { z C : Im z 0 } is not open. Take w = 1 Ω . For every ε > 0 the disk D (1 , ε ) contains the point 1 - i ε/ 2, which does not belong to Ω . This set is in fact closed. 3. Functions, their limits and continuity 3.1. Functions. We have already met functions in real variable theory and in set theory. Recall that every function, f , comes with a domain D ( f ) and a range R ( f ). Then f maps the members of the domain to the members of the range. More precisely, f maps a single member of the domain to a single member of the range. Functions do not map one member of a domain to more than one member of the range.
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