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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116 views

Find $\lim_{n\to\infty} \frac{n^2}{2^n}$

$$\lim_{n\to\infty}\frac{n^2}{2^n}$$ Do you have some tips so I could solve this problem, without the use of L'Hôpital's rule? Indeed, we didn't see formally L'Hôpital's rule, nor Taylor series so ...
1
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3answers
329 views

If $\lim\limits_{x\to\infty} xf(x) = L$, then $\lim\limits_{x\to\infty} f(x) =0$ [duplicate]

Show that if $f: (a,\infty) \rightarrow \mathbb R$ such that $$\lim_{x\to \infty} xf(x) = L$$ where $L \in \mathbb R, $ then $$ \lim_{x\to \infty} f(x) = 0. $$
9
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1answer
160 views

Complete ordered field is an Archimedean field that cannot be extended to an Archimedean field

As a bonus problem, our professor of real analysis asked us to prove that the real numbers (a complete ordered field) cannot be extended into an Archimedean field, with no definition of what he meant ...
2
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3answers
92 views

A simple polynomial differential equation

Consider the differential equation $$ \frac{d}{dx}y(x) = -(y(x))^3. $$ with initial condition $y(0) = 1$. I know that it admits the unique solution $$ y(x) = \sqrt{ \frac{ 1 }{ 1 + 2 x } }. $$ ...
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1answer
93 views

Find all $f: \mathbb{Q} \rightarrow \mathbb{R}$ such that $f(x+y) = f(x)+f(y)$ [duplicate]

i have to find all functions $f: \mathbb{Q} \rightarrow \mathbb{R}$, such that $f(x+y)=f(x)+f(y)$. So functions of the form $f(x) := ax, a \in \mathbb{R}$ satisfy the above condition: $$ ...
0
votes
1answer
148 views

Uniform convergent and lipschitz continuous

I want to prove that if I have a sequence $ f_n\in C[0,1]$ that is uniform convergent to zero and all functions are lipschitz continuous, then the lipschitz constants form a zero sequence. Does ...
1
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1answer
70 views

Does $\int_{0}^{\pi/2n}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t\leq \pi$ hold for $n \geq 2$?

Today I am trying to prove an integral inequality: $$\frac{1}{\pi}\int_{0}^{\pi/2}\left|\frac{\sin(2n+1)t}{\sin t}\right|\text{d}t<\frac{2+\ln n }{2}$$ where $n\geq 2$ and $n \in \Bbb{N}$. First, ...
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0answers
86 views

Clarification of Hölder norm in terms of oscillation

Let $\Omega\subset\mathbb{R}^2$ be an open bounded set, $B(x_0, \rho)=\{x\in\mathbb{R}^2\ |\ |x-x_0|\leq \rho\}$, $\Omega(x_0, \rho)\equiv B(x_0, \rho)\cap \Omega$, $u\in L^{\infty}\big(\Omega(x, ...
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4answers
98 views

How to prove $\sum_{k=1}^n \frac{2^k}{k}< 3\frac{2^n}{n}$?

How to prove $$\sum_{k=1}^n \frac{2^k}{k}< 3\frac{2^n}{n}$$ and further $$\lim_{n\rightarrow \infty}\frac{n}{2^n}\sum_{k=1}^n \frac{2^{k}}{k} = 2$$? These results are verified by computer, yet I ...
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1answer
25 views

about restricting outer measures

If $\mu_0$ is an outer measure on an algebra, we can extend the premeasure to an outer measure $\mu^*$. By Caratheodory's theorem, the collection of $\mu^*$-measurable sets is a $\sigma$-algebra. Is ...
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2answers
221 views

Use mathematical induction to prove that for any $k \in\mathbb N , \lim (1+k/n)^n = e^k$.

Use mathematical induction to prove that for any $k \in \mathbb N, \lim (1+k/n)^n = e^k$. I already used monotone Convergence Theorem to prove $k=1$ case. Do I just need to go through the same ...
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4answers
2k views

What is the relationship between Fourier transformation and Fourier series?

Is there any connection between Fourier transformation of a function and its Fourier series of the function? I only know the formula to find Fourier transformation and to find Fourier coefficients to ...
0
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2answers
65 views

Real Analysis: Covergence Question

Suppose that $( x_n )$ is a sequence of real numbers, $( y_n )$ is a bounded sequence of non-zero real numbers, and that $\lim x_n/ y_n = 1$. Prove that $\lim (x_n - y_n) = 0$. Since $y_n$ is bounded, ...
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0answers
127 views

Showing a sequence $x_{n+1} = Tx_n$ forms a contraction mapping

I want to show that the sequence given by $$x_{n+1} = Tx_n = x_n-\frac{(x_n^2-2)}{x_n+x_{n-1}}$$ forms a contraction mapping. That is $$|Tx_1-Tx_2|\leq c|x_1-x_2|.$$ Where $c$ is to be determined. I ...
2
votes
3answers
103 views

Value of tan(pi/2)

I understand that this is a very stupid question but I'm not getting the answer. At x=pi/2,what s the value of tan(x)?Should it be -infinity or +infinity? Texts tell it to be +infinity.But why? ...
2
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1answer
78 views

Dense subspace of linear functionals

We know that any real measurable function can be approximated by increasing simple functions. So, integral of real valued measurable function can be written as a limit of integrals of simple ...
3
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2answers
632 views

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$

Show that $f(x)=||x||^p, p\ge 1$ is convex function on $\mathbb{R}^n$. I have tried to use Holder's inequality, but I still cannot solve this problem. Could you help me with this problem? Thank you ...
2
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2answers
105 views

Prove that $x_{n+1}=\frac{2}{9}(x_n^3+3)$ converges

Let $x_1=1/2$ and $x_{n+1}=\frac{2}{9}(x_n^3+3)$ for $n\geq 1$. We want to prove that the sequence $(x_n)$ converges to real number $r\in (0,1)$ satisfying the equation $2r^3-9r+6=0$. First part For ...
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2answers
52 views

Continuity of function at $x=0$ [closed]

Discuss continuity of the following function at $x=0$ $$ f(x)=\begin{cases} \pi-x, \text{ for } x < 0 \\ \pi+x, \text{ for } x > 0 \end{cases} $$ Is this function continuous or ...
4
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1answer
140 views

Measure theory singles out the countable cardinal. Why?

In some elementary analysis courses, we discussed what would fail without countable additivity, although it's not as if there would be some contradiction. It would merely be "not nice." We'd lose ...
2
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1answer
56 views

$f^{(n)}(z_0)>n!\, b_n$ and also $\lim (b_n)^{1/ n}\to\infty$

$f$ is analytic, we need to show that it is not possible $f^{(n)}(z_0)>n! b_n\forall n=1,2\dots,\text{ where }\lim (b_n)^{1\over n}\to\infty$ From Taylor Series Expansion around $z_0$ we get ...
7
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1answer
221 views

$\max\min$ problem

Suppose there is a real valued function $f(x,y)$ where $x,y$ are vector variables $x\in\mathbb{R}^n$ $y\in\mathbb{R}^m$. Moreover $f$ is strictly/concave in $x$ and strictly/convex in $y$. $f$ is also ...
3
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4answers
164 views

Let $a_n$ and $b_n$ be sequences of real numbers. If $b_n$ is bounded and $\lim_{n \to \infty} a_n = 0$, then $\lim_{n\to\infty} a_{n}b_{n} = 0$

Prove the below statement: Let $a_n$ and $b_n$ be sequences of real numbers. If $b_n$ is bounded and $\lim_{n \to \infty} a_n = 0$, then $\lim_{n \to \infty} a_n b_n=0$ When I read this question, I ...
2
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2answers
52 views

If $a,b \in \Bbb R$, prove that $|ab| \le (a^2+b^2)/2$

So far I have the first case when $a=b$: \begin{align*} |ab| &= |b^2|\\ &=|b|^2\\ &=\frac{2|b|^2}2\\ &=\frac{b^2+b^2}2\\ &=\frac{a^2+b^2}2 \end{align*} Case 2: $a>b$ Case 3 ...
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1answer
1k views

Absolute continuity of the Lebesgue integral

This is an exercise that I am having trouble with. Not for a grade just for practice. Its an obvious result intuitively but I am having trouble making a rigorous argument. Assume $f$ is Lebesgue ...
2
votes
2answers
85 views

If $|x_{n+1}-x_n| < |x_n-x_{n-1}|$, then $(x_n)$ is a Cauchy sequence

Prove or disprove : If $|x_{n+1}-x_n| < |x_n-x_{n-1}|$ for all $n\geq 2$, then $(x_n)$ is a Cauchy sequence What I understand from this is if the difference between the $n$ and $n+1$ terms in the ...
0
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1answer
38 views

Extract a converging geodesic from a sequence

Let $(X, d)$ be a compact, complete, separable metric space, and $g_n$ a sequence of constant speed geodesic with the same endpoints, i.e. continuous maps $g_n : [0,1] \rightarrow X$ such that $$ ...
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2answers
56 views

proving polynomial root

Prove that $p(x)=6x^4+4x^3-2x^2-x-\pi$ has a root in the interval $[-2,0]$ I was reading an analysis book, and run onto this practice problem, I spent some time already trying to figure out how to do ...
0
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3answers
209 views

Prove that ($x_n$) converges to the cube root of a

i) Let $s,a>0$ and $t=\frac{1}{3}(2s+a/s^2)$, use the Arithmetic-Geometric inequality to prove $t^3\geq a$. Using the Arithmetic-Geometric inequality, I found $$(a_1 a_2 a_3 ... a_n)^{1/n} \leq ...
5
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3answers
280 views

Updated: Constructing a bijection between $\left(-\frac{\pi}{2}, \frac{\pi}{2}\right]$ and $\mathbb{R}$

I am supposed to construct a bijective function for the interval: \begin{align} I_2=\left(-\frac{\pi}{2} ,\frac{\pi}{2} \right] \longrightarrow \mathbb{R} \tag{Problem} \end{align} I first tried the ...
2
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1answer
67 views

Matrix powers sequence bounded

Let $m\in\mathbb{N}^*$ and $A\in\mathcal{M}_m(\mathbb{C})$ such that the matrix sequence $(A^n)_{n\geq 0}$ is bounded. Is the sequence $(\|A\|^n)_{n \geq 0}$ bounded ?
1
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1answer
275 views

Uncountably infinite union/intersection in sigma-algebra

i was wondering whether it makes any difference (or whether it is even true) that an uncountably infinite union/intersection of sets that are elements of a sigma algebra is again an element of the ...
0
votes
1answer
39 views

What's wrong with my argument $f$ is Lipschitz?

Recall the definition: $f: [a,b] \to \mathbb{R}$ is Lipschitz if there is constant $L > 0$ such that $$|f(x)-f(y)| \le L|x-y|, \quad x,\,y \in [a,b]$$ I claim bounded functions are Lipschitz. ...
1
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1answer
46 views

Existence of a point of continuity to the right of any point on a right-continuous function

Let $f: \mathbb{R}\to\mathbb{R}$ be a right-continuous function. How may I show that for any $\epsilon>0$, any $x\in\mathbb{R}$, there exists $y$, $x<y<x+\epsilon$, such that $f$ is ...
0
votes
1answer
49 views

Existence of a geodesic in a complete separable metric space

If I have $X$ a complete separable metric space, $x, y \in X$ arbitrary points, how can I define a constant speed geodesic, i.e. a continuous map $g : [0,1] \rightarrow X$ such that $$ d(g(t), g(s)) = ...
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1answer
88 views

Is the set (0, $\infty$) open?

A set is open if it doesn't contain any of its boundary points. I think 0 is a boundary point here and I think it's the only one. So is the set open?
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1answer
58 views

Monotone Sequence Limit Question

Let $\{a_n\}$ be a monotone increasing sequence that converges to a finite limit. If a monotone subsequence $\{a_{n_k}\}$ (with $n_{k+1}>n_{k}$, and $n_k\rightarrow\infty$) converges to a finite ...
2
votes
1answer
173 views

How prove this $f(x)$ is polynomial function

we define $$f^{[n]}(x)=\lim_{h\to 0}\sum_{k=0}^{n}\binom{n}{k}\dfrac{(-1)^kf[(x+(n-2k)h]}{(2h)^n}$$ if $f(x)$ is continuous on $[a,b]$,and such $f^{[n]}(x)=0$ for all $x$. prove or disprove ...
1
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1answer
81 views

Bochner integration and the associated notion of measurability

In http://en.wikipedia.org/wiki/Bochner_integral a notion of measurability is discussed that depends on the measure $\mu$. Usually measurability does not depend on having a measure anyway. Is this ...
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0answers
41 views

Composition of analytic functions is analytic in a general setting, and are they continuous?

Regarding the notion of analyticity discussed in this setting: A possible equivalence for holomorphicity I wonder if this is truly the correct definition (even though it is from Dunford-Schwarz) An ...
0
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2answers
62 views

If $f$ is unbounded, then is $f$ infinite?

How can this not be? If $n$ is in the image of $f$, then $n+1$ must be in the image of $f$, right? so this tends to infinity?
2
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3answers
203 views

Convergence of alternating nested radicals

Last evening, after reading a couple of questions about nested radicals, I started to wonder about problems involving what I will term "alternating nested radicals;" below is an example, which I found ...
4
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2answers
1k views

Convergence of the arithmetic mean

Let $(a_n)_{n \in \mathbb{N}}$ be a convergent sequence with limit $a \in \mathbb{R}$. Show that the arithmetic mean given by: $$s_n:= \frac{1}{n}\sum_{i=1}^n a_i \tag{A.M.} $$ also converges to $a$. ...
1
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1answer
471 views

Open cover with no finite subcover

Let (x_n) be a sequence, let $L$ ∈ R, and for each ϵ>0, {k ∈ N : x_k ∈ B($L$; ϵ)} Suppose S is not a compact subset of R. There is some ϵ_L > 0, such that {k ∈ N : x_k ∈ B($L$; ϵ_L)} is finite. ...
2
votes
2answers
306 views

Show a Fourier series converges uniformly

I need to show that the Fourier Series of |x| in the interval $(-\pi, \pi)$ converges uniformly to |x| in $[-\pi, \pi]$. I know that |x| = $\frac{\pi}{2}$ + ...
1
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0answers
80 views

Can I use sequential continuity for one sided limits?

First of all sorry if this has been posted before, I had look but couldn't find anything thats exactly like my question. Suggest I am given a function $f[a,b] \rightarrow \mathbb{R} $that is ...
0
votes
1answer
66 views

How to find the closure of a subset

How do I start find the closure of a subset? Let's say I'm given a list, such as $$A=\left\{\frac12,\frac13,\frac23,\frac14,\frac24,\frac34,\frac15,\frac25,\frac35,\frac45,\cdots\right\}$$ using the ...
0
votes
1answer
77 views

How add some condition such $\lim_{u\to+\infty}\int_{a}^{+\infty}f(x,u)dx=\int_{a}^{+\infty}\lim_{u\to+\infty}f(x,u)dx$

Question: if $$\lim_{u\to+\infty}\int_{a}^{+\infty}f(x,u)dx=\int_{a}^{+\infty}\lim_{u\to+\infty}f(x,u)dx\tag{1}$$ My question: then $f(x,u)$ must have Need to add what condition? I know this: ...
0
votes
1answer
73 views

Convergent subsequence

1) Let (x_n) be a sequence and let L ∈ R. Suppose that for each ϵ > 0, {k ∈ N : x_k ∈ B(L; ϵ)} is infinite. Show that (x_n) has a subsequence converging to L. ...
0
votes
1answer
114 views

Limit as x approaches ∞ using epsilon-delta

Let $f$ and $g$ be defined on $(a,\infty)$ and suppose $\displaystyle\lim_{x\rightarrow\infty}(f)=L$ and $\displaystyle\lim_{x\rightarrow\infty}(g)=\infty$. Prove that ...