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Basic analysis: Introduction to real analysis by Jiri Lebl

By Jiri Lebl

A primary direction in mathematical research. Covers the true quantity procedure, sequences and sequence, non-stop services, the spinoff, the Riemann indispensable, sequences of features, and metric areas. initially constructed to educate Math 444 at collage of Illinois at Urbana-Champaign and later better for Math 521 at collage of Wisconsin-Madison. See http://www.jirka.org/ra/

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B + 2B 2 |y| Finally let us tackle (iv). Instead of proving (iv) directly, we prove the following simpler claim: Claim: If {yn } is a convergent sequence such that lim yn = 0 and yn = 0 for all n ∈ N, then 1 1 = . n→∞ yn lim yn lim Once the claim is proved, we take the sequence {1/yn } and multiply it by the sequence {xn } and apply item (iii). Proof of claim: Let ε > 0 be given. Let y := lim yn . Find an M such that for all n ≥ M we have ε |y| |yn − y| < min |y|2 , . 2. FACTS ABOUT LIMITS OF SEQUENCES 51 Now note that |y| = |y − yn + yn | ≤ |y − yn | + |yn | , or in other words |yn | ≥ |y| − |y − yn |.

N→∞ n→∞ Proof. We simply note the reverse triangle inequality |xn | − |x| ≤ |xn − x| . Hence if |xn − x| can be made arbitrarily small, so can |xn | − |x| . Details are left to the reader. 3 Recursively defined sequences Once we know we can interchange limits and algebraic operations, we will actually be able to easily compute the limits for a large class of sequences. One such class are recursively defined sequences. That is sequences where the next number in the sequence computed using a formula from a fixed number of preceding numbers in the sequence.

Let {xn } be a sequence. Let {ni } be a strictly increasing sequence of natural numbers (that is n1 < n2 < n3 < · · · ). The sequence {xni }∞ i=1 is called a subsequence of {xn }. 1. SEQUENCES AND LIMITS 45 For example, take the sequence {1/n}. Then the sequence {1/3n} is a subsequence. To see how these two sequences fit in the definition, take ni := 3i. Note that the numbers in the subsequence must come from the original sequence, so 1, 0, 1/3, 0, 1/5, . . is not a subsequence of {1/n} Note that a tail of a sequence is one type of subsequence.

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