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- Chapter 02 - Sequence and Series
- s// then $\exists$ a positive integer such that $\left| {\,{s_n}}\right|>\frac{1}{2}s$. * Theorem: Let... verge to //s// and //t// respectively. Then (i) $\left\{a{s_n}+b{t_n}\right\}$ converges to $as+bt$. (ii) $\left\{{s_n}{t_n}\right\}$ converges to st. (iii) $\left\{\frac{{{s_n}}}{{{t_n}}} \right\}$ converges to $\fr
- Chapter 01 - Real Number System
- nderline z\in \mathbb{R}^n$ then prove that (a) $\left\| {\,\underline x + \underline y \,} \right\| \le \left\| {\,\underline x \,} \right\| + \left\| {\,\underline y \,} \right\|$ (b) $\left\| {\,\underline x - \underline z \,} \right\| \le \left\| {
- Chapter 03 - Limits and Continuity
- )|<\varepsilon$ whenever //s// and //t// are in $\left\{x:|x-c|<\delta \right\}$. * Theorem (Sandwichi... , $g:f(E)\to Z$ and $h:E\to Z$ defined by $h(x)=g\left(f(x)\right)$. If //f// is continuous at $p\in E$ ... n to $\mathbb{R}^k$ defined by $\underline{f}(x)=\left(f_1(x),f_2(x),f_3(x),...,f_k(x)\right)$, $x\in X$
- Chapter 04 - Differentiation
- //b//) then there exists $x\in (a,b)$ such that $\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$. ==== Download or View