## Real Analysis (Google eBook)Real Analysis builds the theory behind calculus directly from the basic concepts of real numbers, limits, and open and closed sets in $\mathbb{R}^n$. It gives the three characterizations of continuity: via epsilon-delta, sequences, and open sets. It gives the three characterizations of compactness: as ``closed and bounded,'' via sequences, and via open covers. Topics include Fourier series, the Gamma function, metric spaces, and Ascoli's Theorem. The text not only provides efficient proofs, but also shows the student how to come up with them. The excellent exercises come with select solutions in the back. Here is a real analysis text that is short enough for the student to read and understand and complete enough to be the primary text for a serious undergraduate course. Frank Morgan is the author of five books and over one hundred articles on mathematics. He is an inaugural recipient of the Mathematical Association of America's national Haimo award for excellence in teaching. With this book, Morgan has finally brought his famous direct style to an undergraduate real analysis text. |

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Capítulo 17: Sequencias de funções

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accumulation point boundary point bounded interval Cantor set Cauchy sequence Chapter choose closed and bounded closed sets compact set Compute continuous function convergent subsequence converges absolutely converges pointwise Corollary countable sets define definition derivative differentiable domain endpoints Exercises exp(x f is continuous Figure finite subcover fn(x Fourier series fractal function f functions is continuous give a counterexample Give an example given Greek letter Hence infinitely intersection irrational Let f maximum Mean Value Theorem metric space negative terms nonnegative open cover open sets oscillation positive terms power series Proof Proposition Prove or give radius of convergence rationals real analysis real numbers Riemann integral Riemann sums sequence of functions series converges absolutely set contained set is open sets is compact sinx subintervals subsequence converging subset sup metric Suppose that f term by term totally disconnected uncountable uniform convergence uniformly continuous Weierstrass M-test