Merging man & maths https://www.mathcity.org/ Tue, 09 Jun 2026 16:03:06 +0000 FeedCreator 1.8 https://www.mathcity.org/_media/logo.svg https://www.mathcity.org/ https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/real_number_system 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\|$$\underline … anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:59 +0000 https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/sequence_and_series Chapter 02 - Sequence and Series Contents * Sequence, Subsequence, Increasing Sequence, Decreasing Sequence, Monotonic Sequence, Strictly Increasing or Decreasing * Bernoulli’s Inequality * Bounded Sequence * Convergence of the Sequence$s_n<u_n<t_n$$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… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:59 +0000 https://www.mathcity.org/msc/syllabus/pu Syllabus for PU Syllabus and scheme of studies for Regular/Private students doing MSc Mathematics from University of the Punjab, Lahore. 2 years M.Sc Mathematics programme consists of two parts namely Part-I and Part II. The regulation, Syllabi and Courses of Reading for the M.Sc. (Mathematics) Part-I and Part-II (Regular Scheme) are given below. anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:16 +0000 https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/limits_and_continuity 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|… anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:57 +0000 https://www.mathcity.org/msc/syllabus/uos Syllabus for UoS (Private only) Syllabus and scheme of studies for private students doing MSc Mathematics from University of Sargodha, Sargodha. The syllabus has been changed and few optional subjects has been dropped. Please be alert --- 2017/08/25 17:05 anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:50:15 +0000 https://www.mathcity.org/msc/real_analysis_notes_by_syed_gul_shah/differentiation 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|$ anonymous@undisclosed.example.com (Anonymous) Sun, 07 Feb 2021 16:49:57 +0000