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Real Analysis: Short Questions and MCQs @msc:mcqs_short_questions
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n\}$ converges to 5 and $\{y_n\}$ converges to 2. Then find $\lim_{n\to\infty z_n}$, where $z_n=x_n-2y_n... n\}$ converges to 3 and $\{y_n\}$ converges to 4. Then find $\lim_{n\to\infty z_n}$, where $x_n=2y_n-3z_... anel><panel> 6. If a real number is not rational then it is ............... * (A) integer * (B)... hbb{N} \wedge x^2 \leq 7 \} \subset \mathbb{N}$. Then supremum of $A$ is * (A) 7 * (B) 3 *
Chapter 03 - Limits and Continuity @msc:real_analysis_notes_by_syed_gul_shah
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nto //X// (iv) //p// is the limit point of //E//. Then $\lim_{x\to p} f(x)=q$ iff $\lim_{n\to\infty}f(p_... ercies * Theorem: If $\lim_{x\to c}f(x)$ exists then it is unique. * Theorem: Suppose that a real va... open interval //G// except possibly at $c\in G$. Then $\lim_{x\to c}f(x)=l$ if and only if for every po... /G//. If $\lim_{x\to c}f(x)=\lim_{x\to c}g(x)=l$, then $\lim_{x\to c}h(x)=l$. * Theorem: (for sum, dif
Chapter 02 - Sequence and Series @msc:real_analysis_notes_by_syed_gul_shah
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_n\}$ and $\{t_n\}$ converge to same limits as s, then the sequence $\{u_n\}$ also converges to s. * T... rem: If the sequence $\{s_n\}$ converges to //s// then $\exists$ a positive integer such that $\left| {\... {t_n\}$ converge to //s// and //t// respectively. Then (i) $\left\{a{s_n}+b{t_n}\right\}$ converges to $... nce. (i) If $\{s_n\}$ is monotonically increasing then it converges to its supremum. (ii) If $\{s_n\}$ i
Topology: Short Questions and MCQs @msc:mcqs_short_questions
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which is not an open interval. - Let $X=\{a\}$. Then what are the differences between discrete topolog... gy on $X$? - Let $X$ be a non-empty finite set. Then what is the difference between discrete and cofin... Let $\tau$ be a cofinite toplogy on $\mathbb{N}$. Then write any three element of $\tau$. - Let $(\mat... $\tau_2$ are two typologies on non-empty set $X$, then ................... is topological space.\\ -
Chapter 01 - Real Number System @msc:real_analysis_notes_by_syed_gul_shah
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of all positive rationals //p// such that $p^2<2$ then //A// contain no largest member and //B// contain... Let $\underline x,\underline y\in \mathbb{R}^n$. Then (i) $\|\underline x^2\|=\underline x\cdot \underl... ine x,\underline y, \underline z\in \mathbb{R}^n$ then prove that (a) $\left\| {\,\underline x + \under... ion: If //r// is rational and //x// is irrational then prove that $r+x$ and are $rx$ irrational. * Que
Chapter 04 - Differentiation @msc:real_analysis_notes_by_syed_gul_shah
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//f// is differentiable at a point $x\in [a,b]$, then //f// is continuous at //x//. (Differentiability ... imum at a point $x\in [a,b]$ and if $f'(x)$ exist then $f'(x)=0$. (The analogous for local minimum is of... \underline{f}$ be differentiable in (//a//,//b//) then there exists $x\in (a,b)$ such that $\left|\under
Normed Spaces: Short Questions and MCQs @msc:mcqs_short_questions
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on Internet Explorer and does not see the images then follow the instructions given on "); document
Targets @msc:notes
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hink that you have notes to share on MathCity.org then contact to administrator [[:about us|here]]. </WR
Preparation Guide @msc:syllabus:uos
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tic) Note: optional subjects can be chosen other then mentioned above and 100 marks viva will be conduc