Transcription of Chapter 2. Sequences 1. Limits of Sequences
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Chapter 2. Sequences 1. Limits of Sequences
Chapter 2. Sequences 1. Limits of SequencesLetAbe a nonempty set. A function from IN toAis called asequenceof elementsinA. We often use (an)n=1;2;:::to denote a sequence. By this we mean that a functionffrom IN to some setAis given andf(n) =an Aforn IN. More generally, a functionfrom a subset ofZZtoAis also called a is important to distinguish between a sequence and its set of values. The sequence(an)n=1;2;:::given byan= ( 1)nforn IN has infinitely many terms even though theirvalues are repeated over and over. On the other hand, theset{( 1)n:n IN}is exactlythe set{ 1;1}consisting of two sequence (an)n=1;2;:::of real numbers is said toconvergeto the real numberaprovided that for each" >0 there exists a positive integerNsuch that|an a|< "whenevern > N. If (an)n=1;2;:::converges toa, we write limn!1an=a. The numberais called thelimitof the sequence (an)n=1;2.
Chapter 2. Sequences §1.Limits of Sequences Let A be a nonempty set. A function from IN to A is called a sequence of elements in A.We often use (an)n=1;2;::: to denote a sequence.By this we mean that a function f from IN to some set A is given and f(n) …
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