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- Chapter 01 - Real Number System
- ====== Chapter 01 - Real Number System ====== ==== Contents & Summary ==== * Theorem: There is no r... l //p// such that $p^2=2$. * Theorem: Let //A// be the set of all positive rationals //p// such that... uch that $p^2<2$ then //A// contain no largest member and //B// contains no smallest member. * Order on a set. * Ordered set. * Bound. * Least upp
- Chapter 02 - Sequence and Series
- c Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality * Bounded Sequence * Conv... nce * Theorem: A convergent sequence of real number has one and only one limit (i.e. limit of the se... equence * Theorem: A Cauchy sequence of real numbers is bounded. * Divergent Sequence * Theorem:... |>\frac{1}{2}s$. * Theorem: Let //a// and //b// be fixed real numbers if $\{s_n\}$ and $\{t_n\}$ con
- Chapter 03 - Limits and Continuity
- Theorem: Suppose (i) $(X,{d_x})$ and $(Y,{d_y})$ be two metric spaces (ii) $E\subset X$ (iii) $f:E\to... f(x)=l$ if and only if for every positive real number $\varepsilon$, there is $\delta>0$ such that $|f... f metric spaces) * Theorem: Let (i) //X, Y, Z// be metric spaces (ii) $E\subset X$ (iii) $f:E\to Y$,... // is continuous at //p//. * Theorem: Let //f// be defined on //X//. If //f// is continuous at $c\in
- Chapter 04 - Differentiation
- * Derivative of a function * Theorem: Let //f// be defined on [//a//,//b//], if //f// is differentia... Examples * Local Maximum * Theorem: Let //f// be defined on [//a//,//b//], if //f// has a local ma... of vector valued function * Theorem: Let //f// be a continuous mapping of the interval [//a//,//b//... ] into a space $\mathbb{R}^k$ and $\underline{f}$ be differentiable in (//a//,//b//) then there exists