By Charles C. Pugh

Used to be airplane geometry your favourite math direction in highschool? Did you're keen on proving theorems? Are you in poor health of memorizing integrals? if that is so, genuine research can be your cup of tea. unlike calculus and trouble-free algebra, it consists of neither formulation manipulation nor purposes to different fields of technological know-how. None. it's natural arithmetic, and it truly is bound to attract the budding natural mathematician.

In this new creation to undergraduate genuine research the writer takes a distinct strategy from earlier shows of the topic, by way of stressing the significance of images in arithmetic and difficult difficulties. The exposition is casual and cozy, with many useful asides, examples and low reviews from mathematicians akin to Dieudonne, Littlewood and Osserman. This ebook is predicated at the honors model of a direction which the writer has taught repeatedly during the last 35 years at Berkeley. The booklet comprises a good choice of greater than 500 workouts.

**Read Online or Download Real Mathematical Analysis (Undergraduate Texts in Mathematics) PDF**

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**Extra resources for Real Mathematical Analysis (Undergraduate Texts in Mathematics)**

**Example text**

Think also of a function dynamically. At time zero all the elements of A are sitting peacefully in A. Then the function applies itself to them and throws them into B. At time one all the elements that were formerly in A are now transferred into B. Each a E A gets sent to some element f(a) E B. A mapping f : A ---+ B is an injection (or is one-to-one) if for each pair of distinct elements a, a' E A, the elements f(a), f(a') are distinct in B. That is, a f= a' => f(a) f= f(a ' ). The mapping f is a surjection (or is onto) if for each b E B there is at least one a E A such that f(a) = b.

Preserve their meaning. (a) 2 is the smallest prime number. (b) The area of any bounded plane region is bisected by some line parallel to the x-axis. " What makes the following sentence ambiguous? " Negate the following sentences in English using correct mathematical grammar. (a) Ifroses are red, violets are blue. *(b) He will sink unless he swims. Why is the square of an odd integer odd and the square of an even integer even? What is the situation for higher powers? [Hint: Prime factorization.

The cardinal numbers are 0, 1,2, ... The first infinite cardinal number is aleph null, ~o. One says the N has ~O elements. A mystery of math is the Continuum Hypothesis which states that R. has cardinality ~l, the second infinite cardinal. , the Continuum Hypothesis asserts that S ~ Nor S ~ R No intermediate cardinalities exist. You can pursue this issue in Paul Cohen's book, Set Theory and the Continuum Hypothesis. Cardinality Section 4 31 If S is denumerable then there is a bijection f : N --+ S, and this gives a way to list the elements of S as S1 = fO), S2 = f(2), S3 = f(3), etc.