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MTH321: Real Analysis I (Spring 2023)
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onvergent sequence is bounded. - Suppose that $\left\{ {{s}_{n}} \right\}$ and $\left\{ {{t}_{n}} \right\}$ be two convergent sequences such that $\underset... }}$ for all $n\ge {{n}_{0}}$, then the sequence $\left\{ {{u}_{n}} \right\}$ also converges to $s$. - ... irrational number $x$, there exists a sequence $\left\{ {{r}_{n}} \right\}$ of distinct rational number
MTH604: Fixed Point Theory and Applications (Fall 2022)
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$x$ and radius $r$ in some metric space. Find $B\left(0;5 \right)$ in discrete metric on $\mathbb{R}$. ... $x$ and radius $r$ in some metric space. Find $B\left(1;0.7 \right)$ in discrete metric on $\mathbb{R}$... $x$ and radius $r$ in some metric space. Find $B\left(2;5 \right)$ in usual metric on $\mathbb{R}$. -... et $(X, d)$ be a complete metric space and let $B\left(x_0, r\right)$ be open ball with centre at $x_0 \
MTH322: Real Analysis II (Spring 2023)
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epsilon >0\,$there exists a $B>0\,$ such that \[\left| \,\int_{b}^{c}{f\,\,dx}\, \right|<\varepsilon \]... in[a,b]$, there exist an integer $N$ such that $$\left|f_{n+p}(x)-f_n(x) \right| < \varepsilon, \quad n\... , \quad x\in[a,b]$ and let $M_n=\sup_{x\in[a,b]} \left|f_n(x)-f(x) \right|.$ Then $f_n\to f$ uniformly o... positive numbers such that for all $x\in [a,b]$ $\left|f_n(x)\right| \leq M_n \quad \hbox{for all}\,\, n
MTH322: Real Analysis II (Fall 2021)
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there exists a $B>0$ such that $c>b>B$ implies $\left| \,\int\limits_{b}^{c}{f\,\,dx}\, \right|<\vareps... epsilon >0\,$there exists a $B>0\,$ such that \[\left| \,\int_{b}^{c}{f\,\,dx}\, \right|<\varepsilon \]... f(x)$, $x\in[a,b]$ and let $M_n=\sup_{x\in[a,b]} \left|f_n(x)-f(x) \right|$. Then $f_n\to f$ uniformly o... st the for uniform convergence of $\displaystyle \left\{\frac{\sin nx}{\sqrt{n}}\right\}$, $0\leq x\leq
MTH604: Fixed Point Theory and Applications (Spring 2020)
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$x$ and radius $r$ in some metric space. Find $B\left(0;5 \right)$ in discrete metric on $\mathbb{R}$. ... $x$ and radius $r$ in some metric space. Find $B\left(1;0.7 \right)$ in discrete metric on $\mathbb{R}$... $x$ and radius $r$ in some metric space. Find $B\left(2;5 \right)$ in usual metric on $\mathbb{R}$. -... to X$ be a mapping defined by $T(x)=\frac{10}{11}\left(x+\frac{1}{x} \right)$ for all $x\in X$. Prove th
MTH424: Convex Analysis (Fall 2020)
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its interior of its domain. ===Lecture 04=== * Left & right derivative * If $f:I\rightarrow \mathbb... {R}$ and $g:J\rightarrow \mathbb{R}$ where $range\left( f\right) \subseteq J$. If $f\ $and$\ g$ are conv
MATH-510: Topology
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n in $\tau$? - Write the closure of the set $S=\left\{1+\frac{1}{n}: n \in \mathbb{N} \right\}$ in usu
MTH251: Set Topology
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opology on $\mathbb{R}$? * Consider the set $A=\left\{1,\frac{1}{2},\frac{1}{3},... \right\}$. Find th