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MTH322: Real Analysis II (Fall 2021)
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e a$. Assume that $f(x)\ge 0$ for each $x\ge a$. Then prove that $\int_{a}^{\infty }{f(x) dx}$ converg... ge a$ and $\int_{a}^{\infty }{g\,dx}$ converges, then $\int_{a}^{\infty }{f\,dx}$ converges and we hav... $\lim\limits_{x\to \infty }\frac{f(x)}{g(x)}=1$, then $\int\limits_{a}^{\infty }{f dx}$ and $\int\li... that $f\in \mathcal{R}[a,b]$ for every $b\ge a$. Then the integral $\int\limits_{a}^{\infty }{f dx}$ co
MTH321: Real Analysis I (Spring 2023)
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}}<{{u}_{n}}<{{t}_{n}}$ for all $n\ge {{n}_{0}}$, then the sequence $\left\{ {{u}_{n}} \right\}$ also co... olimits_{n=1}^{\infty }{{{a}_{n}}}$ is convergent then $\underset{n\to \infty }{\mathop{\lim }}\,{{a}_{... \ge k$. - If $\sum{{{b}_{n}}}$ is convergent, then $\sum{{{a}_{n}}}$ is convergent. - If $\sum{{{a}_{n}}}$ is divergent, then $\sum{{{b}_{n}}}$ is divergent. - Prove that e
MTH322: Real Analysis II (Spring 2023)
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e a$. Assume that $f(x)\ge 0$ for each $x\ge a$. Then $\int_{\,a}^{\,\infty }{f(x)\,dx}$ converges if,... a$. If $\lim_{x\to \infty } \frac{f(x)}{g(x)}=0$, then convergence of $\int_{a}^{\infty }{g(x)dx}$ impl... c}{f\,\,dx}\, \right|<\varepsilon \] for $b,c>B$, then $\int_{a}^{\infty }{f\,dx}$ is convergent. - If... ts_{a}^{\infty }{f\,dx}$ is absolutely converges, then it is convergent but the converse is not true in
MTH424: Convex Analysis (Fall 2020)
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above. * If $f:[a,b]\to \mathbb{R}$ is convex, then $f$ is bounded above by $\max(f(a),f(b))$. ===... ive * If $f:I\rightarrow \mathbb{R}$ is convex, then $f_{-}'(x)$ and $f_{+}'(x)$ exist and are increas... .$$ * Suppose $f$ is differentiable on $(a,b)$. Then $f$ is convex [strictly convex] if, and only if, ... = * Let $f$ is twice differentiable on $(a,b)$. Then $f$ is convex on $(a,b)$ iff $f''(x)\geq 0$ for
MTH604: Fixed Point Theory and Applications (Fall 2022)
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be a metric space and $F:X\to X$ be a contration, then prove that $\{ F^n(x)\}$ is Cauchy sequence. -... be a metric space and $F:X\to X$ be a contration, then prove that $\{d(F^m(x),F^n(x) \leq \frac{L^n}{1-L... ic space and $T:X\to X$ be a contractive mapping. Then prove that $T$ has unique fixed point. - Let $(... h $d\left(F\left(x_0\right), x_0\right)<(1-L) r$. Then prove that $F$ has a unique fixed point in $B\lef
MTH604: Fixed Point Theory and Applications (Spring 2020)
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... \circ T}_{m\,times}$ is contraction mapping. Then $T$ has a unique fixed point. - Define cover an... ic space and $T:X\to X$ be a contractive mapping. Then prove that $T$ has unique fixed point. - Let $X... d $T:[a,b]\to[a,b]$ be a differentiable function. Then $T$ is a contractive mapping on $X$ if and only i... ined as $T(x)=\frac{1}{1+x^2}$, $x\in[0,\infty)$. Then prove that $T$ is contractive map. - Let $C$ be
MCQs or Short Questions @atiq:sp15-mth321
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real numbers - If a real number is not rational then it is ............... * (A) integer * (B)... et $A=\{x| x\in \mathbb{N} \wedge x^2 \leq 7 \}$. Then supremum of $A$ is * (A) 7 * (B) 3 * ... f finite width. - If the sequence is convergent then * (A) it has two limits. * (B) it is boun... onvergent sequence. $If $\lim_{n\to\infty}s_n=s$, then * (A) $\lim_{n\to\infty}s_{n+1}=s+1$ * (B
MTH321: Real Analysis I (Fall 2021)
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If $r$ is non-zero rational and $x$ is irrational then prove that $r+x$ and $rx$ are irrational. * 1.1... ence $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $... ence $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $... Prove that if $\lim\limits_{n\to\infty}{s_n}=t$, then $\lim\limits_{n\to\infty}{|s_n|}=|t|$ but convers
MTH251: Set Topology
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c,d\}$ and $\tau=\{\varphi, X, \{a\}, \{a,c\}\}$. Then find the relative topology of $A=\{c,d\}$. * Let $A$ be a subset of topological space $X$. Then prove that $A$ is closed in $X$ iff $A'\subset A$. * Let $A$ be a subset of topological space. Then prove that $A\cup A'$ is closed. * Let $A$ be a subset of topological space. Then prove that $\overline{A}=A\cup A'$ * Let $A$ an
MTH321: Real Analysis I (Spring 2020)
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ence $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $... ence $\{s_n\}$ converges to $s$, where $s\neq 0$, then there exists a positive integer $n_1$ such that $... Prove that if $\lim\limits_{n\to\infty}{s_n}=t$, then $\lim\limits_{n\to\infty}{|s_n|}=|t|$ but convers... \infty} y_n=c$. If $x_n<z_n<y_n$ for all $n>n_0$, then the sequence $\{z_n\}$ also converges to $c$. *
MATH-510: Topology
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ou understand the basic knowledge and definitions then there is no problem to answer such type of questi... which is not an open interval. - Let $X=\{a\}$. Then what are the differences between discrete topolog... gy on $X$? - Let $X$ be a non-empty finite set. Then what is the difference between discrete and cofin... Let $\tau$ be a cofinite toplogy on $\mathbb{N}$. Then write any three element of $\tau$. - Let $(\mat
MTH604: Fixed Point Theory and Applications
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ons. * Let $d$ be usual metric on $\mathbb{R}$. Then find open ball, closed ball and sphere with radiu... (F(x),F(y)<d(x,y)$ for $x,y \in X$ and $x\neq y$. Then $F$ has a unique fixed point in $X$. * Let $(X,... _0, r)$ with $0<L<1$) with $d(F(x_0,x_0)<(1-L)r$. Then $F$ has a unique fixed point in $B(x_0,r)$. * L... artial \overline{B_r}) \subseteq \overline{B_r}$. Then $F$ has a unique fixed point in $\overline{B_r}$.
MTH604: Fixed Point Theory and Applications (Spring 2021)
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b{R}$. - Let $F:E\to \mathbb{R}$ be a function. Then prove that $p$ is fixed point of $F$ iff $p$ is t
What is Mathematics? @atiq:math-608
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ms and getting the right answers by right method. Then, since most people lose contact with mathematics