mathAnalysis1-07-draft

mathAnalysis1-07-draft - P r e l i m i n a r y d r a f t o...

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Unformatted text preview: P r e l i m i n a r y d r a f t o n l y : p l e a s e c h e c k f o r fi n a l v e r s i o n ARE211, Fall 2007 LECTURE #1: TUE, AUG 28, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS1) Contents 1. Analysis 1 1.1. References 1 1.2. Sequences 1 1.3. Distance/Metrics 3 1. Analysis Heavy emphasis on proofs in this section. Remember that it was the proofs that killed last years class, and there was no quick way of helping them to do them. So well do a lot of them this semester. The best topic to learn how to proofs in is analysis. 1.1. References Chapter 12 in Simon-Blume Chapter 1 and 2: Elementary Classical Analysis, by J. Marsden (on Reserve) 1.2. Sequences The natural numbers , denoted N , are 1,2,3,4 ..., going on for ever. 1 2 LECTURE #1: TUE, AUG 28, 2007 PRINT DATE: AUGUST 21, 2007 (ANALYSIS1) A sequence is a mapping from the natural numbers to a set S , i.e., f : N S ; f ( n ) is the n th element of the sequence. Typically, we suppress the functional notation: instead of writing the image of n under f as f ( n ) we denote it by x n and write the sequence as { x 1 , x 2 , ..., x n , ... } , i.e., f ( n ) = x n . A sequence { y 1 , y 2 , ..., y n , ... } is a subsequence of another sequence { x 1 , x 2 , ..., x n , ... } if there exists a strictly increasing function : N N such that for all n N , y n = x ( n ) . Note that maps the domain of the subsequence into the domain of the original sequence. For example, consider the sequence { 3 , 6 , ..., 3 n, ... } and the subsequence { 6 , 12 , ..., 6 n, ... } . In this case, the function we need is ( n ) = 2 n , i.e., for all n , y n = x 2 n : e.g., y 1 = x 2 = 6, y 2 = x 4 = 12. That is, you construct a subsequence by throwing out elements of the original sequence, but keeping an infinite number...
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mathAnalysis1-07-draft - P r e l i m i n a r y d r a f t o...

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