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- 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 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... in //Y//. * Theorem: Let $f_1,f_2,f_3,...,f_k$ be real valued functions on a metric space //X// and
- Chapter 02 - Sequence and Series
- c Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality * Bounded Sequence * Conv... |>\frac{1}{2}s$. * Theorem: Let //a// and //b// be fixed real numbers if $\{s_n\}$ and $\{t_n\}$ con... \to \infty}{r_n}=x$. * Let a sequence $\{s_n\}$ be a bounded sequence. (i) If $\{s_n\}$ is monotonic... for all $n>m>n_0$. * Theorem: Let $\sum {a_n}$ be an infinite series of non-negative terms and let
- Chapter 01 - Real Number System
- l //p// such that $p^2=2$. * Theorem: Let //A// be the set of all positive rationals //p// such that... field. * Theorem: (a) Archimedean property (b) Between any two real numbers there exits a rational ... \sqrt{12}$ is irrational. * Question: Let //E// be a non-empty subset of an ordered set, suppose $\alpha$ is a lower bound of //E// and $\beta$ is an upper bound then prove that $\alpha\le \
- 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