Download Basic Real Analysis by Anthony W. Knapp PDF

By Anthony W. Knapp

Systematically increase the techniques and instruments which are important to each mathematician, no matter if natural or utilized, aspiring or established

A entire therapy with an international view of the topic, emphasizing the connections among genuine research and different branches of mathematics

Included all through are many examples and countless numbers of problems, and a separate 55-page part supplies tricks or whole recommendations for most.

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Example text

Under our definitions the intervals of R are the intervals of R∗ that are subsets of R, even if a or b is infinite. If no special mention is made whether an interval lies in R or R∗ , it is usually assumed to lie in R. The next step is to extend the operations of arithmetic to R∗ . It is important not to try to make such operations be everywhere defined, lest the distributive laws fail. Letting r denote any member of R and a and b be any members of R∗ , we make the following new definitions: Multiplication:   +∞ r(+∞) = (+∞)r = 0  −∞   −∞ r(−∞) = (−∞)r = 0  +∞ if r > 0, if r = 0, if r < 0, if r > 0, if r = 0, if r < 0, (+∞)(+∞) = (−∞)(−∞) = +∞, (+∞)(−∞) = (−∞)(+∞) = −∞.

If the sequence is monotone decreasing, the limit is the greatest lower bound of the image. REMARK. 6, rather than the existence of least upper bounds, that is taken for granted in an elementary calculus course. 6 tends for calculus students to be easier to understand than the statement of the least upper bound property. 6. PROOF. Suppose that {an } is monotone increasing and bounded. 1, and let ≤ > 0 be given. If there were no integer N with a N > a − ≤, then a − ≤ would be a smaller upper bound, contradiction.

By the Bolzano–Weierstrass Theorem and the pointwise boundedness, we can find a subsequence of { f n } that is convergent at x1 , a subsequence of the result that is convergent at x2 , a subsequence of the result that is convergent at x3 , and so on. The trouble with this process is that each term of our original sequence may disappear at some stage, and then we are left with no terms that address all the rationals. The trick is to form the subsequence { f n k } of the given { f n } whose k th term is the k th term of the k th subsequence we constructed.

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