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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... tween any two real numbers there exits a rational number. * Theorem: Given two real numbers //x// and //y//, $x<y$ there is an irrational number //u// such that $x<u<y$. * Theorem: For every real number //x// there is a set //E// of rational number suc

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... n$ and $t\ne 0$. * Theorem: For each irrational number //x//, there exists a sequence $\left\{{r_n}\righ... al Principle for Convergence): A sequence of real number is convergent if and only if it is a Cauchy seque... m{a_n}$ is convergent if and only if for any real number $\varepsilon >0$, there exists a positive integer

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