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- Chapter 02 - Sequence and Series
- nded Sequence * Convergence of the Sequence * Theorem: A convergent sequence of real number has one and... the sequence is unique.) * Cauchy Sequence * Theorem: A Cauchy sequence of real numbers is bounded. * Divergent Sequence * Theorem: If $s_n<u_n<t_n$ for all $n\ge n_0$ and if both ... n the sequence $\{u_n\}$ also converges to s. * Theorem: If the sequence $\{s_n\}$ converges to //s// the
- Chapter 03 - Limits and Continuity
- imit of the function, examples and definition * Theorem: Suppose (i) $(X,{d_x})$ and $(Y,{d_y})$ be two m... \to\infty}{p_n}=p$. * Examples and exercies * Theorem: If $\lim_{x\to c}f(x)$ exists then it is unique. * Theorem: Suppose that a real valued function //f// is def... /t// are in $\left\{x:|x-c|<\delta \right\}$. * Theorem (Sandwiching Theorem): Suppose that //f//, //g//
- Chapter 04 - Differentiation
- tiation ====== * Derivative of a function * Theorem: Let //f// be defined on [//a//,//b//], if //f// ... //x//. (Differentiability implies continuity) * Theorem (derivative of sum, product and quotient of two functions) * Theorem (Chain Rule) * Examples * Local Maximum * Theorem: Let //f// be defined on [//a//,//b//], if //f// h
- Chapter 01 - Real Number System
- r System ====== ==== Contents & Summary ==== * Theorem: There is no rational //p// such that $p^2=2$. * Theorem: Let //A// be the set of all positive rationals /... nd (infimum). * Least upper bound property. * Theorem: An ordered set which has the least upper bound p... n ordered field. * Existence of real field. * Theorem: (a) Archimedean property (b) Between any two rea