This is beta site. https://beta.mathcity.org/ 2026-06-07T07:39:58+00:00 https://beta.mathcity.org/ https://beta.mathcity.org/_media/logo.png text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/mcqs_short_questions/real_analysis?rev=1737476041&do=diff Real Analysis: Short Questions and MCQs <callout type=“info” icon=“true”> We are going to add short questions and MCQs for Real Analysis. The subject is similar to calculus but little bit more abstract. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\left\{\frac{1}{n+1} \right\}$$\left\{\frac{n+2}{n+1} \right\}$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$z_n=x_n-2y_n$$\{x_n\}$$\{y_n\}$$\lim_{n\to\infty z_n}$$x_n=2y_n… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/limits_and_continuity?rev=1737476041&do=diff Chapter 03 - Limits and Continuity * Limit of the function, examples and definition * Theorem: Suppose (i) $(X,{d_x})$ and $(Y,{d_y})$ be two metric spaces (ii) $E\subset X$ (iii) $f:E\to Y$ i.e. f maps E into X (iv) p is the limit point of E. Then $\lim_{x\to p} f(x)=q$ iff $\lim_{n\to\infty}f(p_n)=q$ for every sequence {$p_n$} in E such that ${p_n}\ne p$$\lim_{n\to\infty}{p_n}=p$$\lim_{x\to c}f(x)$$c\in G$$\lim_{x\to c}f(x)=l$$\varepsilon$$\delta>0$$|f(t)-f(s)|<\varepsilon$$\left\{x:|x-c|… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/sequence_and_series?rev=1737476041&do=diff Chapter 02 - Sequence and Series Contents * Sequence, Subsequence, Increasing Sequence, Decreasing Sequence, Monotonic Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality * Bounded Sequence * Convergence of the Sequence$s_n<u_n<t_n$$n\ge n_0$$\{s_n\}$$\{t_n\}$$\{u_n\}$$\{s_n\}$$\exists$$\left| {\,{s_n}}\right|>\frac{1}{2}s$$\{s_n\}$$\{t_n\}$$\left\{a{s_n}+b{t_n}\right\}$$as+bt$$\left\{{s_n}{t_n}\right\}$$\left\{\frac{{{s_n}}}{{{t_n}}} \right\}$$\frac{s}{t}$${t_n}\ne… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/mcqs_short_questions/toplogy?rev=1737476041&do=diff Topology: Short Questions and MCQs <callout type=“info” icon=“true”> We are going to add short questions and MCQs for Topology. This is a compulsory subject in MSc and BS Mathematics in most of the universities of Pakistan. The author of this page is Dr. $\mathbb{R}$$X=\{a\}$$X$$X$$X$$\tau$$\mathbb{N}$$\tau$$(\mathbb{Z}, \tau)$$\mathbb{N}$$\tau$$A=\{\pm 100,\pm 101, \pm 102, ... \}$$\tau$$E=\{0,\pm 2,\pm 4,...\}$$\tau$$\tau$$B=\{1,2,3,...,99\}$$\tau$$C=\{10^{10}+n : n \in \mathbb{Z} \}$$\tau$$S… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/real_number_system?rev=1737476041&do=diff Chapter 01 - Real Number System Contents & Summary * Theorem: There is no rational p such that $p^2=2$. * Theorem: Let A be the set of all positive rationals p such that $p^2>2$ and let B consist of all positive rationals p such that $p^2<2$ then A contain no largest member and $x<y$$x<u<y$$x=\sup E$$x>0$$n>0$$y^n=x$$\underline x,\underline y\in \mathbb{R}^n$$\|\underline x^2\|=\underline x\cdot \underline x$$\|\underline x\cdot \underline y\|=\|\underline x\| \|\underline y\|$$\underline … text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/differentiation?rev=1737476041&do=diff Chapter 04 - Differentiation * Derivative of a function * Theorem: Let f be defined on [a,b], if f is differentiable at a point $x\in [a,b]$, then f is continuous at x. (Differentiability implies continuity) * Theorem (derivative of sum, product and quotient of two functions)$x\in [a,b]$$f'(x)$$f'(x)=0$$\mathbb{R}^k$$\underline{f}$$x\in (a,b)$$\left|\underline{f}(b)-\underline{f}(a)\right|\le (b-a)\left|\underline{f'}(x)\right|$ text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/mcqs_short_questions/normed_spaces?rev=1737476041&do=diff Normed Spaces: Short Questions and MCQs <div> <SCRIPT language="JavaScript"> <!-- var browserName=navigator.appName; if (browserName=="Microsoft Internet Explorer") { document.write("If you are viewing this page on Internet Explorer and does not see the images then follow the instructions given on "); document.write("<b><a href=http://w3onlinesource.blogspot.com/2010/08/google-docs-viewer-embed-can-be-seen-in.html target=_blank>this page</a></b>."); } else { document.write("");… text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/notes/targets?rev=1737476041&do=diff Targets Here we have listed the notes for MSc or BS Mathematics, which will be published on MathCity.org. We are working hard to find these notes. Whenever we found these notes we will put them on our website. Here are our targets. * Fluid Mechanics text/html 2025-01-21T16:14:01+00:00 Anonymous (anonymous@undisclosed.example.com) https://beta.mathcity.org/msc/syllabus/uos/preparation_guide?rev=1737476041&do=diff Preparation Guide This guide is made by Mr. Anwar Khan, PhD. We are very thankful to him for sharing. This guide is helpful to prepare papers for MSc Mathematics (annual system) from University of Sargodha. Part 1 1. REAL ANAYSIS * Real Analysis (Notes by Syed Gul Shah) * Chapter # 08 sequences and series of Mathematical Method by SM Yousaf (solutions are available $z= f(x,y)$