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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5answers
274 views

Proof that the set of irrational numbers is dense in reals

I'm being asked to prove that the set of irrational number is dense in the real numbers. While I do understand the general idea of the proof: Given an interval (x,y) choose a positive rational ...
0
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1answer
25 views

What can we say about the definite integral and Riemann sums?

Consider $$f(x) = \int_{1}^{x} \frac{dt}{t}$$ and $$g(x) = \sum_{1}^{x} \frac{1}{t}.$$ I would like to say that $f(x) < g(x)$ for all natural numbers $x \ge 1$. Is there an easy way to derive ...
0
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0answers
20 views

Determine the limit of a series, involving trigonometric functions.

I have $$\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}.$$ I did prove convergence: $$\sum^\infty_{n=1} \frac{\sin((n+1)x)}{(n+1)^3}-\sum^\infty_{n=1} \frac{\sin(nx)}{n^3}≤\sum^\infty_{n=1} ...
1
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1answer
31 views

Matrix representation of shape operator

Let $f$ be a parametrized surface $f: \Omega \subset \mathbb{R}^2 \rightarrow \mathbb{R}^3$ and $N : \Omega \rightarrow Tf$ the Gauß map. Then the shape operator is defined as $L = -DN \circ Df^{-1}.$ ...
0
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1answer
38 views

If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$.

Statement: If $\lim\limits_{z\to z_0} f(z)=0$ and $|g(z)|<M$, for all $z$, with $M$ being a positive number, then we have $\lim\limits_{z\to z_0} f(z)g(z)=0$. I just wanted to verify my proof ...
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0answers
5 views

Riemann Sum of a sequence of partitions proof

Given that $\lim_{||P|| \rightarrow 0} S(P, f) = I$ (where I is in $\Bbb R$ and S(P,f) is the Riemann sum of f). How do I show that if $(P_n)_{n \epsilon N}$ is a sequence of partitions with $||P_n|| ...
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0answers
16 views

Weak star convergence question

Let $C$ be a convex cone in $L^{\infty}$, that is if $x,y \in C$ and $\alpha, \beta > 0 $ then $\alpha x + \beta y \in C$. Let $U$ be the unit ball in $L^{\infty}$. Assume that for each sequence ...
6
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1answer
126 views

Real Analysis : Self Studying vs Doing a Course

I am an engineering graduate student. Recently I got interested in studying Maths. So, I have started self-studying Real Analysis(let's call it RA) using a few books. I will also be using problem ...
1
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2answers
41 views

Jacod Protter “Probability Essentials” Problem 2.8

The question asks to show that a sigma-algebra $\mathcal A$ consisting of $A$ s.t. $A=f^{-1}(B)$, where $B$ is in $\mathcal B$ are Borel subsets of $R$ and $f$ is continuous, is contained in $\mathcal ...
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2answers
63 views

$ \int \frac{dx}{4x^2-12x+13}$

This is probably not too hard but i can't get it right: I am trying to calculate $$\displaystyle \int \frac{dx}{4x^2-12x+13}$$. The solution is $\displaystyle ...
1
vote
1answer
34 views

Question in analysis: subset of open interval in $\Bbb R$

Consider metric space $(X,d)$, $X=(a,b)\subset \Bbb R$, $d(x,y)= \lvert x-y \rvert$. Let a subset $S \subset (a,b)$ be open and closed. Show that either $S=(a,b)$ or $S= \emptyset$. There's a ...
2
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1answer
106 views

A question about Heine-Borel Theorem.

Im reading chapter2 of rudin's Principle of Mathematical analysis. Heine-Borel theorem is involved in this chapter, $\mathbf{2.41}\,\,$ Theorem$\,\,\,$ If a set $E$ in $R^k$ has one of the ...
1
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1answer
18 views

Application of the IVT

Is it true that on any circle there is a pair of opposite points where the age of the surface rock is the same? I think the answer is no. In the temperature case the function T: [ 0, 2π] → R where ...
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0answers
9 views

Does $-\Delta u\equiv u^p$ have non-positive radial solutions?

Let $p>1$ and $u:[0,R)\to\mathbb{R}$ be a radial solution of $$\left\{\begin{matrix}\displaystyle-u''-\frac{n-1}ru'&\equiv&u^p&&\text{on }(0,R)\\ u'&\equiv ...
1
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1answer
12 views

Proof via strong induction of a string output

I'm still new to the whole proof thing (first class of discrete mathematics and analysis right now). I could do general induction problems, but the fact that 'n' is the output here along with the ...
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0answers
15 views

Inclusion of Sobolev spaces with fractional order

Let $W^{k,p}\mathbb(R^n)$ the usual Sobolev space. We know that if $k>l$ and $1\leq p<q<\infty$, $(k-l)p<n$ and $$\frac{1}{q}=\frac{1}{p}-\frac{k-l}{n}$$ then ...
2
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3answers
52 views

the choice of 2 when proving the limit when $x\to\infty$

Suppose that $f$ is a continuous function on $\mathbb{R}$ and $\lim_{x\to -\infty}f(x)$ and $\lim_{x\to -\infty}f'(x)$ exist. Show that $\lim_{x\to -\infty}f'(x)=0$ A common way to show this is ...
1
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1answer
26 views

Proof of the second principle of mathematical induction

This was an exercise in my lecture notes for which no answer was provided, so I seek verification on whether my proof is correct. Prove that if 1. $P(n_0)$ is true for some $n_0 \in \mathbb N$, and ...
0
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1answer
21 views

>>A. $X$ banch. >>B.M banach. >>C. $X/M$ banach.

I want to prove that any two will imply the other: A. $X$ banch. B.M banach. C. $X/M$ banach. I have proved A+B imply C by taking M closed in X. I am not understanding ...
2
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1answer
38 views

Weyl asymptotic law

In Panoramic view in Riemannian geometry of Berger, I met the following formula $$\sum e^{-\lambda_i t} \sim \frac{\vert \Omega\vert }{2\pi t} -\frac{\vert \partial \Omega\vert}{\sqrt{2\pi t}} + ...
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1answer
17 views

$\lim_{n \to \infty} \int_{\Omega}X_n d \mu = +\infty$ under some conditions

Suppose $X_n$ are measurable functions in $L^1$ defined on the measure space $(\Omega, \mathfrak{F}, \mu)$. Suppose that $0 \leq X_n$ a.e. for all $n$ and $X_n \leq X_{n+1}$ a.e. for all $n$. Thus ...
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0answers
21 views

Show that $\int_a^b f(x) dx=\lim_{n\rightarrow \infty} \sum_{k=0}^{n-1} \int_{x_k}^{x_{k+1}} f(x) dx$.

I've come up with a proof for the following statement, but I'm not quite sure it's 100% correct. I would appreciate any help: If $f$ is integrable on $[a,b]$, $x_0=a$, and $x_n$ is a sequence of ...
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0answers
13 views

Variants of the change-of-variables formula

Consider the following change of variables formula for $f:X\rightarrow Y$, that holds for any "reasonable" $g:B\subseteq Y \rightarrow \mathbb{R}$ and $A\subseteq X$ $$ \int_B g(x)\ {\rm ...
3
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1answer
51 views

False equations with Euler's Identity [duplicate]

What's wrong with the following equations? $$1 = 1^{-i} = (e^{2πi})^{-i} = e^{-i2πi} = e^{2π}$$ My guess would be the third equation, but I can't really tell why... in the first equation, we use the ...
0
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1answer
65 views
+50

Isometric and conformal map

We defined conformal and isometric maps for surfaces $f,g: \Omega \subset \mathbb{R}^2 \rightarrow S \subset \mathbb{R}^3$. Under a reparametrization of $f$ I understand a diffeomorphism $\Phi : M ...
0
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0answers
47 views

Product rule in fractional Sobolev spaces

I have to prove the following inequality $$\Vert fg\Vert_{H^s(\mathbb{R}^3)}\leq C \Vert f\Vert_{H^\sigma(\mathbb{R}^3)}\Vert g\Vert_{H^{s+1}(\mathbb{R}^3)}$$ with $s\geq 0$, ...
2
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1answer
60 views

uniform continuity and equivalent sequences

Let $X$ be a subset of $\mathbb{R}$, and let $f : X\to \mathbb{R}$ be a function. Then the following two statements are logically equivalent: (a) $f$ is uniformly continuous on $X$. (b) ...
5
votes
1answer
31 views

Difficulty with Jensen's Equation.

Its easy to find all continuous function $f: \Bbb R \to \Bbb R $ satisfing the Jensen equation $$f \left( \frac{x+y}{2}\right )=\frac{f(x)+f(y)}{2}$$ But I am finding difficulty in finding all ...
5
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1answer
95 views
+100

Intuition behind functional dependence

What is the intuition behind functional independence ? (This is defined in the following way: Let $k\leq n$. The $C^1$ functions $F_1,\ldots,F_k:\mathbb{R}^n\rightarrow \mathbb{R}$ are functionally ...
2
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4answers
54 views

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$.

Let $\{a_n\}$ be a sequence with limit $\alpha$, and define $b_n=a_{n+1}$ where $n\in \mathbb{N}$. Show that $\{b_n\}\rightarrow \alpha$. What I have: Since $\{a_n\}\rightarrow \alpha$ we know that ...
0
votes
1answer
28 views

Bounding summations

Show that $\sum k2^k = \Theta( k2^k)$. I tried to use mathematical induction to prove the bound, but it didn't work. There are other ways that can be used to prove this bound, like bounding the ...
0
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3answers
73 views

Proof of an inequality by induction

Let $n \in \mathbb N^+$. Show that if $x_1, x_2, ... , x_n$ are $n$ real numbers such that $-1 \le x_i \le 0$ for each $1 \le i \le n$, then $$(1 + x_1)(1 + x_2)...(1 + x_n) \ge 1 + x_1 + x_2 + ... + ...
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0answers
21 views

Lebesgue integral of cartesian product of functions

Given two Lebesgue Integrable functions $f,g$, is there a notion of the integral $$\int_A f \times g \, \, dx_1 \times dx_2 ?$$ Is this even a definable notion? I couldn't find anything on the ...
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0answers
27 views

Multiplication on Reals as equivalence classes of cauchy sequences is well defined

So I understand the solution for the proof that multiplication of equivalence classes of cauchy sequences is well defined using boundedness of cauchy sequences and a chain of inequalities. I just ...
0
votes
2answers
27 views

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$.

Let $\{b_n\}$ be a sequence with limit $\beta$. Show that if $B$ is an upper bound for $\{b_n\}$, then $\beta \leq B$. What I have: Assume that $\beta>B$, so $\beta-B>0$. Since $\{b_n\}$ ...
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0answers
78 views

Explain about absolute convergence and convergence [on hold]

$$\sum_{n=1}^{\infty}\frac{{{(-1)^{n}\sin (n)}}}{{{n}}} $$ Explain about absolutely convergent and convergent. Thx a lot.
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0answers
36 views

Inflection point and 2nd derivative

Is it possible a function $f:\mathbb{R} \rightarrow \mathbb{R}$ to have an inflection point somewhere but that it is not two times differentiable at that point? If so, then can we have a form of that ...
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0answers
13 views

Role of functional equations in current panorama of pure mathematics

It seems that currently functional equations are greatly explored as a research field. I would like to know what is the importance and role of such a field in the panorama of the current development ...
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2answers
44 views

How to prove the closure $\bigcup_{n\ge 1} F_{n}$ is totally bounded and closed

Let $(X,p)$ be a metric space. Write $F$ for the set of subsets of $X$ which are closed, bounded, and non-empty. For each integer $n\geq1$, write $F_{n}$ for the set of subsets of X which are finite ...
2
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1answer
25 views

Looking for example of a surjective homomorphism on $(\mathbb R,+)$ which is not an automorphism

Give example of a surjective function $f:\mathbb R \to \mathbb R$ such that $f(x+y)=f(x)+f(y) , \forall x,y \in \mathbb R$ but $f$ is not injective . I think I have to do something with basis of ...
1
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1answer
42 views

Let $f$ and $g$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $|f(1)-f(0)| \le g(1)-g(0)$

Let $f:[0,1] \rightarrow \mathbb{R}^m $ and $g:[0,1] \rightarrow \mathbb{R}$ differentiables such that $|f'(t)| \le g'(t)$, for all $t \in [0,1].$ Prove that $$|f(1)-f(0)| \le g(1)-g(0)$$ Comments ...
2
votes
2answers
77 views

How do I solve this Olympiad question with floor functions?

Emmy is playing with a calculator. She enters an integer, and takes its square root. Then she repeats the process with the integer part of the answer. After the third repetition, the integer part ...
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0answers
54 views

Terrence Tao, Analysis 1. Exercise 5.3.2. Real Numbers and Cauchy Sequences.

Let $ x = \lim_{n\rightarrow\infty}a_n, y = \lim_{n\rightarrow\infty}b_n$, and $ x' = \lim_{n\rightarrow\infty}a'_n$ be real numbers. Then $xy$ is also a real number. Furthermore, is $x=x'$, then $xy ...
4
votes
1answer
29 views

If $a_n$ and $b_n$ are equivalent sequences and $a_n$ is bounded then so is $b_n$.

This is what i know; If $(a_n)$ is an infinite sequence of which is bounded then we can say; $|a_i| < M $ for all $i \geq 0.$ since $a_n$ and $b_n$ are equivalent sequences, we can say that for ...
1
vote
1answer
30 views

prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$

The real function ${g}$ is continuous on $[0,1]$ .we define ${f_n}$ on ${[0,1]}$: $$f_n(x)=\frac{{{g(x)\sin^{n} (x)}}}{{{1+nx}}}$$ prove $({f_n})_n$ is uniformly convergent on ${[0,1]}$ .
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2answers
56 views

Prove that $(f_n)_n$ is uniformly convergent.

Let $g$: $[0,1]\to\mathbb{R}$ be continuous and $g({1})=0$. Define $f_n(x)= x^{n}{g(x)}$. Prove that $(f_n)_n$ is uniformly convergent.
-1
votes
1answer
42 views

absolutely convergent & conditionally convergent [on hold]

Prove that $$\sum_{n=1}^{\infty}\left(\sum_{m=1}^{n}\frac{{{1}}}{{{m}}}\right)\frac{{{\sin(nx)}}}{{{n}}} $$ for $x = {k\pi}$ , $k\in \mathbb{Z} $ is absolutely convergent. & for $x ...
-1
votes
0answers
17 views

Exponential estimate/inequality

I have a vector $x=(x_1,\dots, x_n)\in \mathbb{R}^n$ and some variance $\sigma^2 >0$. I know that the following inequality is wrong (but I present it because it would make world nicer in my view) ...
0
votes
2answers
57 views

Prove$\sum_{n=0}^{\infty} \frac{n}{(n+1)^2}-\frac{1}{n+2} $ is convergent [on hold]

prove $$\sum_{n=0}^{\infty}\frac{{{n}}}{{{(n+1)^{2}}}}-\frac{{{1}}}{{{(n+2)}}} $$ is convergent.
0
votes
2answers
43 views

$ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent. [on hold]

prove that if $ \sum_{n=1}^{\infty} {a_n} $ is absolutely convergent $ \sum_{n=1}^{\infty} \frac{{{n+1}}}{{{n}}}{a_n} $ is absolutely convergent.