Proving real polynomials of degree greater than or equal to 3 are reducible
this is the proof I'm given:Example 1.20. Let $f \in \mathbb{R}[x]$ and suppose that $\deg(f) > 3$. Then $f$ is reducible.Proof. By the Fundamental theorem of algebra there are $A \in C$ such that...
View ArticleIf $P(x,y)$ and $Q(x,y)$ are $\mathcal{C}^1$ functions such that $\oint_C...
Is the following correct?Let $P(x,y),\, Q(x,y)$ be $\mathcal{C}^1$ functions in $\mathbb{R}^2$ such that $\oint_C Pdx+Qdy=0$ for every circle $C$ in $\mathbb{R}^2$Suppose that there is a point...
View ArticleTheorem: Sequence is convergent if an only if that is Cauchy's sequence
I have a problem with proof, I can prove in a one way$$ \epsilon > 0\\|a_n- a| < \epsilon/2\\|a_m- a_n| = |(a_m- a) - (a_n- a)| \leq |a_m- a| + |a_n- a| < \epsilon $$I dont know in a other way.
View Article$|a-b|
For $a,b\in\Bbb R$ prove $|a-b|<E$ iff $a-E<b<a+E$.I am unsure how to start this proof. I think it might have something to do with the reverse triangle inequality but I am stuck.
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$\eta= f(y,z) dy \wedge dz + g(x,z)dz \wedge dx +z^2(x^2+y^2) dx \wedge dy$....
$f, g : \mathbb{R}^2 \to \mathbb{R}$ are functions of class $C^1$. We describe a differential 2-form:$$\eta = f(y, z) dy \wedge dz \ + \ g(x, z) dz \wedge dx \ + \ z^2 (x^2 + y^2) dx \wedge...
View ArticleBernoullis inequality proof
Hi I'm asked to do a proof for bernoullis inequality which is$(1+a)^n \geq 1+na$ where $a\geq-1$I'm proving by induction by the way.So far these are my steps$(1+a)^1 \geq 1+a$Then$(1+a)^k \geq...
View ArticleIs there a number that is always greater than $\log 2^{2n}$ for $n \geq 1$?...
I am trying to write a radius of a disk that is always greater than or equal to $\log 2^{2n}$ for $n \geq 1.$ but has no dependence on n.Can anyone help me in this idea please?
View Article$U\in \mathbb{R}^2$ bounded, smooth edge, $0\in U$, $f\in C_0^{\infty}(U)$....
$U \in \mathbb{R}^2$ is a bounded region with a smooth edge, such that $0 \in U$$f \in C_0^{\infty}(U)$$V(x) = \begin{cases} \frac{x}{||x||^2} \ \text{for:} \ x \neq 0 \\ 0 \ \text{for:} \ x = 0...
View ArticleDoubt in proof of differentiability of $\begin{equation} f(x,y)=...
Let $f$ be:\begin{equation}f(x,y)= \begin{cases} \frac{x^2y^2}{x^2+y^4}, (x,y) \neq (0,0) \\ 0, (x,y)=(0,0) \end{cases}\,\end{equation}Supposedly this is differentiable in (0,0).But I think if we do...
View ArticleGeometric meaning of Archimedean property of real numbers.
I was reading about real number system briefly as a part of my pre-calculus study and read about Archimedean property of real numbers which says thatIf $ x \gt 0 $ and if * y * is an arbitrary real...
View ArticleLet $A$ be a nonempty bounded subset of $\mathbb R$ and let $B$ be the set of...
Let $A$ be a nonempty bounded subset of $\mathbb R$ and let $B$ be the set of all upper bounds for $A$. Prove that $\sup A= \inf B$. Can someone please help me? I'm very confused as to what to do.
View ArticleFinding sequence in a set $A$ that tends to $\sup A$
I have been reading the book at http://www.neunhaeuserer.de/short.pdf, and have noticed that in the proof of the intermediate value theorem (Theorem 5.8 in the book), it seems to be quietly assumed...
View ArticleIf two subsequences converge to different values, prove directly the sequence...
Let $\{x_n\}$ be a sequence. Suppose that there are two convergent subsequences $\{x_{n_{\Large{i}}} \}$ and $\{x_{m_{\Large{i}}} \}$. Suppose that $\lim\limits_{i\to\infty} x_{n_{\Large{i}}} = a$ and...
View ArticleProve that for all $x\in\Bbb R$, precisely one of the statements $x>0$,...
I came across the following exercise while studying Terrence Tao's book Analysis I:Exercise 5.4.1 Let $x\in\Bbb R$. Show that precisely one of the statements $x>0$, $x=0$, or $x<0$ holdsMy...
View ArticleFinding a limit of a monotonically decreasing function
Let $a_n$ be a series so $a_1=1$ and $2a_{n+1}<a_n<3a_{n+1}$. Find the limit of $a_n$ when $n\to \infty$.I have proved that $a_n>0$ and that it monotonically decreasing. But how I find the...
View ArticleShowing that $\mathbb{Q}$ is not complete
Show that there is no least upper bound for $A=\{x: x^2<2\}$ in $\mathbb{Q}$. Suppose $\alpha \in \mathbb{Q}$ is the least upper bound of $A$. Then either $\alpha^2 < 2$ or $\alpha^2 > 2$. I...
View ArticleProving that a function defined as an expectation is Lipschitz
I would like to prove thisLet $G(y)\doteq\mathbb{E}[X|\gamma X+\gamma^{1/2}Z=y]$ where $Z\sim\mathcal{N }(0,1)$ and $X\sim\pi(x)$ where $\pi(x)$ has bounded support. $Z$ and $X$ are indepenedent. I...
View ArticleProve that $f(x) = 0$ for all $x \in [a,b]$, where $f$ is differentiable on...
Q. Suppose $f$ is differentiable on $[a,b]$, $f(a) = 0$ and there is real number $A$ such that $|f'(x)| \leq A|f(x)|$ on $[a,b]$. Prove that $f(x) = 0$ for all $x \in [a,b]$.My attempt: if $A=0$, it is...
View ArticleProof of convexity of $f(x)=x^2$
I know that a function is convex if the following inequality is true:$$\lambda f(x_1) + (1-\lambda)f(x_2) \ge f(\lambda x_1 + (1-\lambda)x_2)$$ for $\lambda \in [0, 1]$ and $f(\cdot)$ is defined on...
View ArticleThe set of numbers whose decimal expansions contain only 4 and 7
Let $S$ be the set of numbers in $X=[0,1]$ that when expanded as a decimal form, the numbers are 4 or 7 only.The following are the problems.a), Is S countable ?b), Is it dense in $X$ ?c), Is it compact...
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