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- Real Analysis Notes by Prof Syed Gul Shah
- s// then $\exists$ a positive integer such that $\left| {\,{s_n}}\right|>\frac{1}{2}s$. * Theorem: L... 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 $\fr
- Metric Spaces (Notes)
- )$ be a metric space. Then for any $x,y\in X$, $$\left| {\,d(x,\,A)\, - \,d(y,\,A)\,} \right|\,\, \le \,... nd only if $(x_n)$ has a convergent subsequence $\left(x_{n_k}\right)$ which converges to $x\in X$. * ... $ converges to $x\in X$, then every subsequence $\left(x_{n_k}\right)$ also converges to $x\in X$. * T... , b] is complete. * Theorem: If $(X,d_1)$ and $\left(Y,d_2\right)$ are complete then $X\times Y$ is co
- Notes for Numerical Methods by M Usman Hamid
- integration * Trapezoidal rule * Simpson’s $\left(\frac{1}{3}\right)$ rule * Simpson’s $\left(\frac{3}{8}\right)$ rule * Weddle’s * Boole’s rule
- Ring (Notes) by Prof. M. Dabeer Mughal
- kernel of $\phi$ and related theorems * Ideal, left ideal * Right ideal, two sided ideal, examples