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MTH321: Real Analysis I (Spring 2023)
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is bounded. - Suppose that $\left\{ {{s}_{n}} \right\}$ and $\left\{ {{t}_{n}} \right\}$ be two convergent sequences such that $\underset{n\to \infty }{\math... {{n}_{0}}$, then the sequence $\left\{ {{u}_{n}} \right\}$ also converges to $s$. - For each irrational... $x$, there exists a sequence $\left\{ {{r}_{n}} \right\}$ of distinct rational numbers such that $\unde
MTH604: Fixed Point Theory and Applications (Fall 2022)
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adius $r$ in some metric space. Find $B\left(0;5 \right)$ in discrete metric on $\mathbb{R}$. - Let $B(... ius $r$ in some metric space. Find $B\left(1;0.7 \right)$ in discrete metric on $\mathbb{R}$. - Let $B(... adius $r$ in some metric space. Find $B\left(2;5 \right)$ in usual metric on $\mathbb{R}$. - Define: Li... be a complete metric space and let $B\left(x_0, r\right)$ be open ball with centre at $x_0 \in X$ and rad
MTH322: Real Analysis II (Spring 2023)
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,$ such that \[\left| \,\int_{b}^{c}{f\,\,dx}\, \right|<\varepsilon \] for $b,c>B$, then $\int_{a}^{\inf... integer $N$ such that $$\left|f_{n+p}(x)-f_n(x) \right| < \varepsilon, \quad n\geq N, p\geq 1 \hbox{ and... and let $M_n=\sup_{x\in[a,b]} \left|f_n(x)-f(x) \right|.$ Then $f_n\to f$ uniformly on $[a,b]$ if and on... bers such that for all $x\in [a,b]$ $\left|f_n(x)\right| \leq M_n \quad \hbox{for all}\,\, n.$ - Let $\
MTH322: Real Analysis II (Fall 2021)
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implies $\left| \,\int\limits_{b}^{c}{f\,\,dx}\, \right|<\varepsilon $. - Suppose $f\in \mathcal{R}[a,b... ,$ such that \[\left| \,\int_{b}^{c}{f\,\,dx}\, \right|<\varepsilon \] for $b,c>B$, then $\int_{a}^{\inf... and let $M_n=\sup_{x\in[a,b]} \left|f_n(x)-f(x) \right|$. Then $f_n\to f$ uniformly on $[a,b]$ if and on... of $\displaystyle \left\{\frac{\sin nx}{\sqrt{n}}\right\}$, $0\leq x\leq 2\pi$. - Test the for uniform
MTH604: Fixed Point Theory and Applications (Spring 2020)
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adius $r$ in some metric space. Find $B\left(0;5 \right)$ in discrete metric on $\mathbb{R}$. - Let $B(... ius $r$ in some metric space. Find $B\left(1;0.7 \right)$ in discrete metric on $\mathbb{R}$. - Let $B(... adius $r$ in some metric space. Find $B\left(2;5 \right)$ in usual metric on $\mathbb{R}$. - Define Lip... efined by $T(x)=\frac{10}{11}\left(x+\frac{1}{x} \right)$ for all $x\in X$. Prove that $T$ is a contracti
MTH424: Convex Analysis (Fall 2020)
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erior of its domain. ===Lecture 04=== * Left & right derivative * If $f:I\rightarrow \mathbb{R}$ is ... $g:J\rightarrow \mathbb{R}$ where $range\left( f\right) \subseteq J$. If $f\ $and$\ g$ are convex and $g
What is Mathematics? @atiq:math-608
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ulas and impossible word problems and getting the right answers by right method. Then, since most people lose contact with mathematics after high school or afte
MATH-510: Topology
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he set $S=\left\{1+\frac{1}{n}: n \in \mathbb{N} \right\}$ in usual topology on $\mathbb{R}$. - What is
MTH251: Set Topology
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the set $A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$. Find the derive set of $A$ under usual topolo