# Tagged Questions

Mathematical analysis. Consider a more specific tag instead: (real-analysis), (complex-analysis), (functional-analysis), (fourier-analysis), (measure-theory), (calculus-of-variations), etc. For data analysis, use (data-analysis).

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### unboundness of an infinite series $f(t)\cos(tx)\sim t^{-1}\cos(tx)$

If $\lim_{t\to \infty} f(t)t=1$, i.e., $f(t)\sim t^{-1}$, then $${\text{ess}\sup}_{x\in [-\pi,\pi]}\sum_{t=1}^{\infty} f(t)\cos(tx)=\infty ?$$ Here ${\text{ess}\sup}$ is ...
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### How to prove if $\int^{\infty}_{0}f(x)dx$ a converges, then there is increasing sequence $x_n$, $\lim_{n \to \infty}f(x_n)=0$

I tried to prove it directly, but examples like $\sin(x^{2})$ makes it impossible to find the proper subsequence $x_{n}$; I also tried proving by contraposition, but the converse negative statement ...
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### Metric space of non empty closed bounded parts of $R$ with the Hausdorff metric

Consider the metric space of non empty closed bounded parts of $R$ with the Hausdorff metric. For n $\in N_{0}$ and $F_{n} = \{0,1/n,2/n,3/n, ..., 1\}$ i am wondering if $(F_{n})_{n}$ is convergent? ...
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### Analysis and algebra [closed]

I'd like to know if there exist a field of the theoretical math that really combines analysis and algebra. Some people say that Model theory combines those two subjects but I personally want ...
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### Controlling the size of a function

Consider the function $$f(\delta,r)=\frac{2e^{-\delta r} }{r}\sinh\left(r/2\right)$$ with $\delta >0$ Show that $\exists r >0 \text{ such that }f(\delta,r)<1$ My attempt: \begin{align*} ...
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### $f(x)=\frac{x}{x}$ is continuous at $0$?

$x$ is divided by $x$. Thus, $f(x)=1$ when $x \neq 0$. However, at $0$ can we consider $f(x)$ as $1$? More specifically, do we have to define a rational function as a reduced form?
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### Any Video Lectures Of An MIT, Harvard, Stanford, UC Berkeley, Yale, or Princeton Analysis Course Based On Baby Rudin?

I'm learning analysis from the book Principles of Mathematical Analysis by Walter Rudin, third edition. This book, popularly known as Baby Rudin, is being used for analysis courses at such elite ...
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### Showing that $f$ is convex given that $(\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), \;\;\forall x,y \in \mathbb{R}^n$

Assume that $f$ is continuously differentiable and that for some constant $c > 0,$ the gradient $\nabla f$ satisfies, (\nabla f(x) - \nabla f(y)) \cdot (x-y) \geq c(x-y)\cdot(x-y), ...
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### Estimate the number of candidates who obtained fewer than 70 scores.

In an examination, the number of candidates who obtained scores between certain limits are as follows: Scores $0—19$, $20—39$, $40—59$, $60—79$, $80—99$, Number of candidates $41$, $62$, $65$, ...
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### Show that the limit $\displaystyle \lim_{n\to \infty }\frac{a_{n}}{n}$ exists.

So $\left \{ a_{n} \right \}_{n\geq 1}$ is a sequence of real numbers and $C>0$ is a fixed constant. We assume that $a_{n+m}\leq a_{n}+a_{m}+C, \forall n, m\geq 1$. What is a good way to prove ...
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### Limit over general chain

We all know classic definition of limit of a sequence. There is also definition of limit of a function. Now consider general chain, i.e. linearly ordered set $(\mathcal{I}, \le)$ with specified values ...
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### The space of sequences which are eventually zero in $l^2$ is not a Hilbert space.

Define $V$ to be the space of sequences which are eventually zero, i.e. $$V=\bigcup_{N=1}^{\infty}\{(x_n)_{n=1}^{\infty}\in l^2: x_n=0 \; \text{for}\; n\ge N\}.$$ Is $V$ a Hilbert space with ...
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### Prove a doubly periodic entire analytic function in complex plane is a constant [duplicate]

I got stuck on this problem. So I really appreciate if anyone can give me some hint to move on. Thanks a lot. Prove that an entire analytic function $f:\mathbb{C} \rightarrow \mathbb{C}$ is a ...
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### Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$.

Let $f$ be analytic in $D(0,2)$. Assume that for all $n∈\mathbb{N}$ $\int_{ |z|=1} {f(z)\over(n+1)z−1}dz=0$. Prove that $f(z)=0$ for all $z∈D(0,2)$. I'm thinking about a contradiction proof. Assuming ...
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### Aproximation in $W^{1,p}(U)$ with U disconnected.

Consider $U=(-1,0)\cup(0,1)$. Define $$v(x)=\left\{\begin{array}{rc} 0,&\mbox{se}\quad -1<x<0,\\1, &\mbox{se}\quad 0<x<1. \end{array}\right.$$ Clearly $v\in W^{1,p}(U)$ for each ...
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### $\sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1$

I can't prove that the following inequality true or not: $$\sup\limits_{t>0}[\frac{g(t)}{\sup\limits_{t<u<\infty}g(u)}]\leq 1 \tag{*},$$ where $g$ is a positive function. I think it is true ...
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### Theorem 3.19 in Baby Rudin: The upper and lower limits of a majorised sequence cannot exceed those of the majorising one

Here is Theorem 3.19 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Let $\{s_n \}$ and $\{t_n \}$ be sequences of real numbers. If $s_n \leq t_n$ for $n \geq N$, ...
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### Theorem 3.17 in Baby Rudin: Infinite Limits and Upper and Lower Limits of Real Sequences

Here're Definitions 3.15 and 3.16 and Theorem 3.17 in the book Principles of Mathematical Analysis by Walter Rudin, 3rd edition. Definition 3.15: Let $\{s_n \}$ be a sequence of real numbers ...
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### About $L^{p}$ norms and the Hilbert transform
When we are proving $L^{p}$ estimates for the Hilbert transform, we can proceed in the following way: Step 1 Prove that $H$ maps $L^{2}$ to $L^{2}$. Step 2 Prove that $H$ maps $L^{1}$ to ...
### Proof of $ax\le bx, x>0 \Rightarrow a \le b$ using only field axioms
I'm reading ELEMENTARY CLASSICAL ANALYSIS (2nd, Marsden), 1.1.2 Proposition. How can I prove $ax\le bx \land x>0 \implies a \le b$ using only sixteen field axioms? My proof was Prove \$\forall ...