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Real Analysis: Short Questions and MCQs @msc:mcqs_short_questions
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uences, whose sum is convergent. - Prove that $\left\{\frac{1}{n+1} \right\}$ is decreasing sequence. - Is the sequence $\left\{\frac{n+2}{n+1} \right\}$ is increasing or decre... em is not interval. * (A) $(1,2)$ * (B) $\left(\frac{1}{2},\frac{1}{3} \right)$ * (C) $[3. \... ed.$</collapse> </panel> <panel> 5. A sequence $\left\{\dfrac{1}{n} \right\}$ is * (A) bounded.
Chapter 02 - Sequence and Series @msc:real_analysis_notes_by_syed_gul_shah
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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 $\fr
Chapter 01 - Real Number System @msc:real_analysis_notes_by_syed_gul_shah
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nderline z\in \mathbb{R}^n$ then prove that (a) $\left\| {\,\underline x + \underline y \,} \right\| \le \left\| {\,\underline x \,} \right\| + \left\| {\,\underline y \,} \right\|$ (b) $\left\| {\,\underline x - \underline z \,} \right\| \le \left\| {
Chapter 03 - Limits and Continuity @msc:real_analysis_notes_by_syed_gul_shah
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)|<\varepsilon$ 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 @msc:real_analysis_notes_by_syed_gul_shah
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//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
Topology: Short Questions and MCQs @msc:mcqs_short_questions
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n in $\tau$? - Write the closure of the set $S=\left\{1+\frac{1}{n}: n \in \mathbb{N} \right\}$ in usu