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204Section22009

# 204Section22009 - Econ 204 Section 2 GSI Hui Zheng Key...

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Econ 204 Section 2 GSI: Hui Zheng Key Words Deduction, Contraposition, Contradiction, Induction, Binary Relation, Equivalence Relation, Equivalence Class, Cardinality, Bijection, Numerically Equivalent Section 2.1 Methods of Proof ° Deduction To prove A ) Z , deduction goes like A ) B ) ± ± ± ) Y ) Z ° Contraposition To prove A ) Z , contraposition is to prove : Z ) : A: It goes like : Z ) : Y ) ± ± ± ) : B ) : A ° Induction A typical structure of proof is For n = 0 (or other initial value), show that the statement is true. This is the base step. For n = k; suppose that the statement is true. This is the inductive hypothesis. For n = k + 1 ; use what we get from the inductive hypothesis to show that the statement holds for the case of n = k + 1 Conclude that the statement is true for all n . ° Contradiction To prove A ) Z by contradiction, we °rst suppose Z is not true. Then we check whether it leads to results that contradict with A or what we get from A . Contraposition can be regarded as a special case of contradiction. Contraposition means : Z ) : A so that we get results that contradicts with A . ° De°nition Convergence of a Sequence of Real Numbers in Euclidean Metric Space. A sequence of real numbers f x n g converges to a real number x if 8 " > 0 9 N ( " ) 2 N , for all n > N ( " ) ) j x n ² x j < " . We denote it as x n ! x or lim n !1 x n = x: The following two examples will use this de°nition. We are going to learn the general de°n- ition of convergence of sequences in metric spaces in Lecture 3. However, they are pretty similar. Example 2.1.1 Prove the following statement by deduction: f x n g is a sequence of real numbers. If lim n !1 x n = x > 0 ; then there exists N 2 N such that n > N ) x n > 0 .

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204Section22009 - Econ 204 Section 2 GSI Hui Zheng Key...

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