Transcription of Math 401 - Introduction to Real Analysis
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Math 401 - Introduction to Real Analysis
math 401 - Introduction to real AnalysisTopics for Midterm I - Review1 - BijectionsA mapf:A7 Bis aninjectionif it is one-to-one, distinct elementsa1, a2 Ahavedistinct imagesf(a1)6=f(a2). The mapfis asurjectionif it is onto, every elementb Bisthe image of some element say that a mapf:A7 Bis abijectionif it isone-to-oneandonto. If a bijectionexists, we regard the two setsAandBas having the same number of elements. This allows us tocompare also sets with infinitely many - Mathematical inductionGiven a sequence of statementsP1, P2, P3, .., mathematical induction is a technique for prov-ing that all of the statements are true. Namely, one has to show that(i) The first statementP1is true.(ii) IfPkis true, then also the following statementPk+1is - Upper bound, supremumA setS IRisbounded aboveif there exists a numberusuch thatu xfor allx this caseuis called anupper bound. The smallest upper bound is calledsupremumandwritten (completeness of the real numbers).
Math 401 - Introduction to Real Analysis Topics for Midterm I - Review 1 - Bijections A map f : A → B is an injection if it is one-to-one, i.e. distinct elements a1,a2 ∈ A have distinct images f(a1) 6= f(a2).The map f is a surjection if it is onto, i.e. every element b ∈ B is
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