By Curtiss D.R.
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Extra resources for Analytic functions of a complex variable
Also, note that the hypothesis that y satisfy 0 < y < 1 is not necessary, but it is the only interesting situation as the conclusion of part (b) is trivially satisfied for y > 1. 32 1 The Real Numbers We now turn our attention to the rational numbers. This set has the same cardinality as the natural numbers but they are more interspersed throughout the real number system than the natural numbers and the integers. The following definition gives a concrete meaning to this intuitive statement. 7. Let S be a subset of R.
A/. We now seek to show that x 2 D 2. To do this, we will rule out the possibilities x 2 < 2 and x 2 > 2. First suppose that x 2 < 2. We will produce a real number t such that x < t and 2 t < 2. This will show that t 2 A but will contradict the fact that x is the supremum of A. To choose the point t we will add a tiny increment to x, the motivation for which is obtained from the inequality x 2 < 2. From our assumption, we know that 2 x 2 > 0, and we set t DxC x2 2 4 : It is clear that x < t since the second term in the definition of t is positive; the factor of 1=4 is chosen so as to keep t close to x in order to guarantee that t 2 < 2.
To show that A and C have the same cardinality, we must exhibit a bijective function from A to C . This of course will depend on the fact that we can construct bijections between A and B, and between B and C ; this is requested in the exercises. To classify the cardinalities of sets we have the following definition. 15. B/ where B D f1; 2; : : : ; ng. The empty set has cardinality 0. N/ or if A is finite, and uncountable otherwise. The definition indicates that the natural numbers form the basis for the smallest type of infinite sets.