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The result of two products of negligible functions

According to the definitions of negligible functions, the product of a negligible function and a positive polynomial results in a negligible function. With this in mind, I was also wondering whether...

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Tangent line of integral

We know that if $F(x) = \int_2^x g(t)dt$, then the slope of the tangent line to $F$ at a point $x_0$ is $$F’(x_0) = g(x_0)$$ by fundamental theorem of calculus.Now suppose that we want to find the...

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Find the area of a figure bounded by curves

Find the area of a figure bounded by curves$$\frac{x^2}{a^2}+\frac{y^2}{b^2}=1,x^2+y^2=ab,x^2+y^2\geq ab,a>b$$I called $x^2=ab-y^2$ and substituted the ellipse into the equation, which gave me the...

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prove that f is injective

Let$ bv_0 = \{(x_k)\in c_0 |\sum_{i=1}^\infty |x_{k+1} -x_k| < \infty \}$$l_1 := \{ (x_k)\in c_0 |\sum_{i=1}^\infty |x_k| < \infty \} $We define$ f: l_1 \to bv_0 $$ f((x_k))= (x_{k+1} -x_k )$I...

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Proving that a sequence converges, with the epsilon definition

I want to prove that the following sequence $x_n = \frac{3+\sqrt{n}}{2n-\sqrt{n}}$ converges and has a limit.$$\lim_{n \to \infty} \frac{3+\sqrt{n}}{2n-\sqrt{n}} \Rightarrow \lim_{n \to \infty}...

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Fourier transform $\mathcal{F}$ of smooth functions $f\in C^l$ decays fast.

I want to prove the above statement, which motivates why the Fourier transform on the Schwartz space is automorphic.Let $f\in C^l(\mathbb{R}^d)$ and $\alpha$ a multiindex of at most order $l$. From the...

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Accumulation points of the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor$

I'm trying to find the accumulation points for the sequence $c_n = \lfloor \cos(\sqrt{n}) \rfloor, n \in \mathbb{N}_0$. I know what the points are, but I'm having trouble coming up with an explicit...

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$x_{1} = \sqrt{2}$ and $x_{n+1} = \sqrt{2+x_{n}}$. Show that the sequence...

The question$x_{1} = \sqrt{2}$ and $x_{n+1} = \sqrt{2+x_{n}}$. Show that the sequence converges and find it's limit.AttemptIf I assume that the limit exists, then I am able to find the limit. But how...

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$f_n \xrightarrow{d} f$ if and only if $f_n \xrightarrow{m} f$ in...

Let $L^0([0,1])$ be the vector space of Lebesgue-measurable functions on $[0,1]$. Let $d$ be the metric on $L^0([0,1])$ given by $$d(f,g) = \int_0^1 \frac{|f(x)-g(x)|}{1 + |f(x)-g(x)|}\, dx.$$Prove...

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How do I prove this function is bounded from below?

$A \in \mathcal{M}_{n,n}(\mathbb{R})$ a positive definite matrix, $b \in \mathbb{R}^n$ and $c \in \mathbb{R}. $$f : \mathbb{R}^n \to \mathbb{R}$ defined by : $$f(x) = \frac{1}{2}\langle Ax, x \rangle +...

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Given x=u²+8 and dy/du=15u³, find dy/dx in terms of x [closed]

Differentiation. Finding dy/dx in terms of x.

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Proof of Du Bois-Reymond Lemma using Riesz representation theorem

I’m working with this version of the fundamental calculus of variations lemma:If $f\in L^p(\mathbb{R}^n)$ and $\int f\phi dx = 0 $ for all $\phi \in C^\infty_c(\mathbb{R}^n)$, then $f=0$ a.e.My...

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"arbitrarily close" and "sufficiently close" confusion in a limit definition

After $2$ minutes $22$ seconds on this video, Dr. Linda Green said on an informal definition of the limit of a function thatFor any function $f(x)$ and for real numbers $a$ and $L$, $\lim_{x\to a} f(x)...

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"arbitrarily close" and "sufficiently close" confusion in a limit definition

After $2$ minutes $22$ seconds on this video, Dr. Linda Green said on an informal definition of the limit of a function thatFor any function $f(x)$ and for real numbers $a$ and $L$, $\lim_{x\to a} f(x)...

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Existence of inf(A)

In my analysis textbook, the definition of a "complete" set is as follows: Let $S$ be an ordered field. Then $S$ is complete if $sup(A) \in S$ for every nonempty $A \subseteq S$ where $A$ is bounded...

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We let P ⊂ R. given t = inf P and if we know g : R->R is increasing and cont....

Let $P \subset\mathbb{R}$. Given $t = \inf P$ and $g : \mathbb{R}\to\mathbb{R}$ increasing and continuous at $t$, how can one show that $g(P)$ is bounded below and $g(t) = \inf g(P)$?

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Consider a cdf $F:[0,1]\to [0,1]$ such that $F(x)\leq x$ for all $x\in...

As the title says, I am considering a CDF $F:[0,1]\to [0,1]$ such that $F(x)\leq x$ for all $x\in [0,1]$.This seems like a pretty basic property and I was wondering if:There is a name for the CDFs that...

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A functional equation with functions of class $C^{1}(\mathbb{R})$

Find all functions $f \in C^{1}(\mathbb{R})$ such that$$f(x)^{2} = \int_{0}^{x} (f(t)^{2}+f'(t)^{2}) dt + 2022$$for all $x$ real.My attempt: If $f \in C^{1}(\mathbb{R})$ is a solution, then$$2f(x)f'(x)...

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Show that $\displaystyle\lim_{n \to \infty} \prod_{k=1}^{n}...

I have a question regarding limit calculations. As part of something I'm working on, I faced the challenge of calculating the limit $$\lim_{n \to \infty}\prod_{k=1}^{n} \left(1-2^{k-1-n}\right).$$This...

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Finding independent copies of converging random variables

Suppose that $X_i \sim X_j$ and independent. Morever suppose there exists $X_{n,i} \rightarrow X_i$ as $n \rightarrow \infty$ a.s and $\mathrm{supp}(X_{n,i}) \subset \mathrm{supp} X_i$ a.s. Does there...

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