Basic Real AnalysisIdeal for the one-semester undergraduate course, Basic Real Analysis is intended for students who have recently completed a traditional calculus course and proves the basic theorems of Single Variable Calculus in a simple and accessible manner. It gradually builds upon key material as to not overwhelm students beginning the course and becomes more rigorous as they progresses. Optional appendices on sets and functions, countable and uncountable sets, and point set topology are included for those instructors who wish include these topics in their course. The author includes hints throughout the text to help students solve challenging problems. An online instructor's solutions manual is also available. |
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absolutely convergent an+1 Archimedean Property Bolzano-Weierstrass Theorem Calculus Cauchy sequence Cauchy's choose continuous function convergent iff converges uniformly Corollary definition derivative differentiable divergent element Example Extreme Value Theorem f(xn fn(x formula function f(x geometric Hence Hint I₁ increasing sequence infinite number least upper bound Lemma Let f(x lim f(x lim sn lim sup lim x→0 liman lower bound Mathematics Mean Value Theorem Monotone Sequence Theorem natural numbers neighborhood nth term open interval open set partial sums partition polynomial positive integer positive number power series Proof Let Proof See Problem Prove directly Prove Theorem Radius of Convergence Ratio Test rational numbers Real Analysis real numbers Rolle's Theorem sequence of functions sequence sn sequential criterion series converges Show Sn(x strictly increasing subset supremum Taylor's Theorem tends to zero Theorem 11 uniformly continuous xn+1 π₁ πη