By George R. Exner

For scholars Congratulations! you're approximately to take a path in mathematical evidence. while you're anxious in regards to the complete factor, this e-book is for you (if no longer, please learn the second one and 3rd paragraphs within the advent for professors following this, so that you will not think left out). The rumors are real; a primary direction in facts can be quite demanding since you must do 3 issues which are most likely new to you: 1. learn arithmetic independently. 2. comprehend proofs by yourself. :1. observe and write your individual proofs. This publication is all approximately what to do if this checklist is threatening since you "never learn your calculus ebook" or "can't do proofs. " this is the excellent news: you want to be sturdy at arithmetic otherwise you shouldn't have gotten this some distance. this is the undesirable information: what labored ahead of won't paintings this time. luck may well lie in bettering or discarding many conduct that have been more than enough as soon as yet usually are not now. let's have a look at how we've got gotten to some extent at which a person might dare to indicate that you've undesirable behavior. l the common undemanding and highschool arithmetic schooling within the usa has a tendency to educate scholars to have useless studying behavior, 1 within the first paragraph, but. xiv creation and we blush to confess collage might be simply as bad.

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D E ~ FE. These rules of thumb may sound reasonable. As a practical student concentrating on doing some mathematics, though, three questions are fair: Do they help me read proofs? Do they help me discover proofs? Do they help me write up proofs I have discovered? The next section is all about the answer to the first of these and also casts some light on another aspect of these rules. 2 Real-Life Proofs vs. Rules of Thumb The following is an example of the sort of proof one might find in a mathematics textbook at the junior level.

102. 103. 105. 9 Summary and More Propaganda The message of this chapter is that there is a way to help you learn mathematics independently and actively, namely to search for and construct examples. There are several classes of examples: basic examples, to get you started; extended examples, to help find the boundaries of the theorem or definition; non-examples, to help separate conditions and find relationships among them. There are standard places to look for examples, among them small and extreme ones.

Prove that this sequence converges. 1. Note also that, for each n, we have by an easy computation that Sn :::; tn· to 1, n-l 1+ L k=O 1 2k ' n = 1,2, .... 3. To show (Sn)~=o is bounded above, we employ an auxiliary sequence (tn)~=o· 4. Therefore, by the theorem mentioned above, we have gent. (sn)~=o conver- 5. Observe that (tn)~=o is convergent, since it is simply the sum of the constant sequence whose value is one and a familiar sequence that is the sequence of partial sums of a geometric series.

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