User 温泽海

5 votes

Why can we simplify $(x^2-4)/(x-2)$ to get $x+2,$ but what happens at $x=2$ when $y=(x^2-4)/(x-2)?$

answered Jun 15, 2022 at 0:09

4 votes

Say that $x \notin A$ or $x \in B$..

answered Jun 14, 2022 at 20:05

4 votes

Accepted

Prove this by induction - how to?

answered Apr 28, 2022 at 15:18

4 votes

How is $\dfrac{4x}{x-3}$ equal to $4+\dfrac {12}{x-3}$?

answered May 3, 2022 at 14:24

3 votes

Books for upper-undergraduate, higher level real analysis

answered Jun 17, 2022 at 13:59

3 votes

Accepted

Find solution of $(f(x))^3 – 3f(x) + x = 0$

answered Jun 19, 2022 at 11:51

3 votes

If $A \subsetneq B$ and $B \subsetneq C$, then $A\subsetneq C$.

answered Jul 24, 2023 at 20:54

3 votes

Finding correlation Between $Z$ and $Y$ Given $\mathrm{Z} | \mathrm{X} = x \sim N(x, \kappa^2)$ and $(\mathrm{X}, \mathrm{Y}) \sim N$

answered Jul 28 at 13:59

2 votes

Accepted

If $\lim\limits_{x\to +\infty} f(x)=L, (L\in \mathbb{R})$, $f'(x)$ is uniformly continuous on $(0,+\infty),$ then $\lim\limits_{x\to +\infty} f'(x)=0$

answered Jun 12 at 12:54

2 votes

Topological game on $(0,1)$

answered May 1 at 14:15

2 votes

$[(p \to q) \land (q \to r)] \to (p \to r)$ is a tautology

answered Mar 1, 2023 at 0:41

2 votes

Accepted

Do there exist $g_n(x)$ such that $\int |g_n| \to 0$ but $\liminf |g_n(x)| \gt 0 \ \forall x \in (0,1)$?

answered May 3, 2023 at 19:24

2 votes

Is it true that $\min\{h, f_n\}\to h$ in measure provided $f_n\to f$ in measure?

answered May 10, 2023 at 2:40

2 votes

Accepted

if $\mu(E)=0$ then $\int_E f d\mu=0$, even if $f(x)= \infty$ for all x in E

answered Jul 24, 2023 at 1:27

2 votes

Accepted

How to prove whether the integrals of two equivalent infinitesimal functions are also equivalent infinitesimal in certain cases

answered Aug 12 at 13:58

2 votes

Let V be a vector space with basis $e_1,...,e_n$. Is $e_1 + e_2,e_2 + e_3,e_3 + e_4,...,e_{n−1} + e_n,e_n + e_1$ a basis of this space?

answered Jun 26, 2022 at 1:23

2 votes

Accepted

Do they mean parallel instead of perpendicular?

answered Jul 8, 2022 at 1:09

2 votes

Accepted

Why is $|m(\xi)| < C_k\left(1 + \sum_{|\xi_j| < R^\delta} \xi_j^2\right)^{-\frac{k}{2}}R^{\delta k}$?

answered Jun 17, 2022 at 3:17

2 votes

Is there a name for the operation on vectors $A$ and $B$ that yields $\|A\|\|B\|\cos\theta$?

answered May 5, 2022 at 0:05

2 votes

Let $g$ be continuous on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one. Show $F$ is either empty or uncountable.

answered Mar 14, 2022 at 16:20

2 votes

Accepted

Calculate $1+2+\ldots+(i-1)+(i-1)+(i+1)+(i+2)+\ldots+(j-1)+(j-1)+(j+1)+(j+2)+\ldots+n $

answered Mar 12, 2022 at 1:30

2 votes

Finding stable sets from a graph

answered Nov 27, 2021 at 16:21

2 votes

Accepted

Is $\lfloor{\frac{a+b+c+d}{4}}\rfloor=\lfloor\frac{\lfloor{\frac{a+b}{2}}\rfloor+\lfloor{\frac{c+d}{2}}\rfloor}{2}\rfloor$ for $a,b,c,d\in\mathbb R$?

answered Jan 7, 2022 at 17:25

2 votes

Accepted

Suppose $n\geq 3$ and let $k$ be $n$ or $n-1$, whichever is odd. Show that the set of $k$-cycles in $A_n$ is not a conjugacy class in $A_n$.

answered Jan 8, 2022 at 1:54

2 votes

Accepted

Understanding the Proof of the existence of the Inverse function of a multivariable function.

answered Jan 12, 2022 at 2:05

2 votes

Show $\sum_{n=2}^{\infty}\frac{1}{n(\log n)^{p}}$ is convergent.

answered Jan 17, 2022 at 3:20

1 vote

Accepted

$K$ be compact metric space, $g$ and $h$ are continuous, prove the family of $f_u(x) = h(g(x)+g(u))$ is equicontinuous.

answered Mar 12, 2022 at 1:20

1 vote

How is the distance between the n-th term of a converging sequence and its' limit not zero?

answered Nov 28, 2021 at 19:01

1 vote

How to prove product and quotient of smooth functions is smooth

answered Mar 12, 2022 at 1:40

1 vote

Accepted

Sequence of continuous functions $f_n $ with $\lim_{n \rightarrow +\infty} f_n(x)=f(x)$ and property

answered Mar 13, 2022 at 15:13

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68 Answers

Why can we simplify $(x^2-4)/(x-2)$ to get $x+2,$ but what happens at $x=2$ when $y=(x^2-4)/(x-2)?$

Say that $x \notin A$ or $x \in B$..

Prove this by induction - how to?

How is $\dfrac{4x}{x-3}$ equal to $4+\dfrac {12}{x-3}$?

Books for upper-undergraduate, higher level real analysis

Find solution of $(f(x))^3 – 3f(x) + x = 0$

If $A \subsetneq B$ and $B \subsetneq C$, then $A\subsetneq C$.

Finding correlation Between $Z$ and $Y$ Given $\mathrm{Z} | \mathrm{X} = x \sim N(x, \kappa^2)$ and $(\mathrm{X}, \mathrm{Y}) \sim N$

If $\lim\limits_{x\to +\infty} f(x)=L, (L\in \mathbb{R})$, $f'(x)$ is uniformly continuous on $(0,+\infty),$ then $\lim\limits_{x\to +\infty} f'(x)=0$

Topological game on $(0,1)$

$[(p \to q) \land (q \to r)] \to (p \to r)$ is a tautology

Do there exist $g_n(x)$ such that $\int |g_n| \to 0$ but $\liminf |g_n(x)| \gt 0 \ \forall x \in (0,1)$?

Is it true that $\min\{h, f_n\}\to h$ in measure provided $f_n\to f$ in measure?

if $\mu(E)=0$ then $\int_E f d\mu=0$, even if $f(x)= \infty$ for all x in E

How to prove whether the integrals of two equivalent infinitesimal functions are also equivalent infinitesimal in certain cases

Let V be a vector space with basis $e_1,...,e_n$. Is $e_1 + e_2,e_2 + e_3,e_3 + e_4,...,e_{n−1} + e_n,e_n + e_1$ a basis of this space?

Do they mean parallel instead of perpendicular?

Why is $|m(\xi)| < C_k\left(1 + \sum_{|\xi_j| < R^\delta} \xi_j^2\right)^{-\frac{k}{2}}R^{\delta k}$?

Is there a name for the operation on vectors $A$ and $B$ that yields $\|A\|\|B\|\cos\theta$?

Let $g$ be continuous on an interval $A$ and let $F$ be the set of points where $g$ fails to be one-to-one. Show $F$ is either empty or uncountable.

Calculate $1+2+\ldots+(i-1)+(i-1)+(i+1)+(i+2)+\ldots+(j-1)+(j-1)+(j+1)+(j+2)+\ldots+n $

Finding stable sets from a graph

Is $\lfloor{\frac{a+b+c+d}{4}}\rfloor=\lfloor\frac{\lfloor{\frac{a+b}{2}}\rfloor+\lfloor{\frac{c+d}{2}}\rfloor}{2}\rfloor$ for $a,b,c,d\in\mathbb R$?

Suppose $n\geq 3$ and let $k$ be $n$ or $n-1$, whichever is odd. Show that the set of $k$-cycles in $A_n$ is not a conjugacy class in $A_n$.

Understanding the Proof of the existence of the Inverse function of a multivariable function.

Show $\sum_{n=2}^{\infty}\frac{1}{n(\log n)^{p}}$ is convergent.

$K$ be compact metric space, $g$ and $h$ are continuous, prove the family of $f_u(x) = h(g(x)+g(u))$ is equicontinuous.

How is the distance between the n-th term of a converging sequence and its' limit not zero?

How to prove product and quotient of smooth functions is smooth

Sequence of continuous functions $f_n $ with $\lim_{n \rightarrow +\infty} f_n(x)=f(x)$ and property