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- Real Analysis Notes by Prof Syed Gul Shah
- s$ a positive integer such that $\left| {\,{s_n}}\right|>\frac{1}{2}s$. * Theorem: Let //a// and //b/... /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 $\frac{s}{t}$ provided ${t_n}\ne
- Metric Spaces (Notes)
- ,y\in X$, $$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,\,d(x,\,y).$$ * Diameter of a set ... x_n)$ has a convergent subsequence $\left(x_{n_k}\right)$ which converges to $x\in X$. * (ii) If $(x_n)... o $x\in X$, then every subsequence $\left(x_{n_k}\right)$ also converges to $x\in X$. * Theorem: Let (/... r any sequence $(x_n)$ in //M//, ${x_n}\to x\quad\Rightarrow\quad x\in M$, then //M// is closed. * Comp
- Notes for Numerical Methods by M Usman Hamid
- Trapezoidal rule * Simpson’s $\left(\frac{1}{3}\right)$ rule * Simpson’s $\left(\frac{3}{8}\right)$ rule * Weddle’s * Boole’s rule * Rectangular rule
- Handwritten Notes of Real Analysis by Asim Marwat
- t test. * Chapter 03: Limit of a Function * Right hand limit, signum function, intermediate value t
- Ring (Notes) by Prof. M. Dabeer Mughal
- and related theorems * Ideal, left ideal * Right ideal, two sided ideal, examples and theorems *