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Chapter 01 - Real Number System
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====== Chapter 01 - Real Number System ====== ==== Contents & Summary ==== * Theorem: There is no ratio... property. * Field. * Proofs of axioms of real numbers. * Ordered field. * Theorems on ordered fiel... (a) Archimedean property (b) Between any two real numbers there exits a rational number. * Theorem: Given two real numbers //x// and //y//, $x<y$ there is an ir
Chapter 02 - Sequence and Series
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quence * Theorem: A convergent sequence of real number has one and only one limit (i.e. limit of the seq... y Sequence * Theorem: A Cauchy sequence of real numbers is bounded. * Divergent Sequence * Theorem: ... $. * Theorem: Let //a// and //b// be fixed real numbers if $\{s_n\}$ and $\{t_n\}$ converge to //s// and... n$ and $t\ne 0$. * Theorem: For each irrational number //x//, there exists a sequence $\left\{{r_n}\righ
Chapter 03 - Limits and Continuity
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c}f(x)=l$ if and only if for every positive real number $\varepsilon$, there is $\delta>0$ such that $|f(... f// is continuous at $c\in X$ then there exists a number $\delta>0$ such that //f// is bounded on the open... n [//a//,//b//] and $f(a)\neq f(b)$, then given a number $\lambda$ that lies between $f(a)$ and $f(b)$, th