Tagged Questions

Theoretical foundations of calculus: limits, convergence of sequences, construction of the real numbers, least upper bound property, and related analysis topics such as continuity, differentiation, integration through the fundamental theorem of calculus.

learn more… | top users | synonyms

1
vote
1answer
26 views

Prove there exists at least $n$ zeros of $f$ in the open interval $(a,b)$

Suppose that $f\in [a,b]$, and $\int_a^b f(x)x^k\,dx=0$, $\quad k=0$, $1$, $\ldots$, $n-1$. Show that there exists at least $n$ zeros of $f$ in the open interval $(a,b)$. I know that if $n=1$, it is ...
2
votes
1answer
30 views

Convergence of $x_{n+1} = x_n + \dfrac{(\vert x_n \vert)^{1/2}}{n^2}$

Let $(x_n)$ be the sequence defined by $x_{n+1} = x_n + \dfrac{(\vert x_n \vert)^{1/2}}{n^2}$ for $n \geq 2$ and $x_1$ be any real number. Then I want to prove that $x_n$ is convergent. It is ...
0
votes
0answers
23 views

A basic real analysis question

consider the following iteration $$x_{n+1} = x_n + a(n)(E[h(y_n) || \mathscr{F_{n-1}}] - x_n)$$ with $\sum a(n) = \infty$ and $\sum a(n)^2 \lt \infty$ Also, $\sup_n|E[h(y_n) || \mathscr{F_{n-1}}]| ...
5
votes
1answer
44 views

Can a power series converge uniformly on $(-1,1)$ but not on $[-1,1]?$

I am taking a course in analysis, and I am wondering whether it possible for a power series with radius of convergence $1$ to converge uniformly on $(-1,1)$ but not on $[-1,1]?$ I don't think this ...
1
vote
3answers
30 views

Weakly monotonic sequences

I am confused with the definition of 'Weakly Monotonic Sequences'. Which one is correct? A sequence which is neither increasing nor decreasing A sequence which is not increasing or not decreasing ...
1
vote
1answer
25 views

Convergent Sequences

Is the sequence $\{$cos$(\pi\sqrt{n^2+n})\}_{n=1}^\infty$ convergent? I guess it is divergent because cos function is oscillating. But not sure. I am stuck in doing justification too. Any ideas?
1
vote
2answers
32 views

Functions that get infinitely “tight”

I was thinking recently about what happens to certain periodic functions as their period decreases. Take as an example, the following function $$ \varphi(x) = \begin{cases} \{x\}, & 0 \le \{x\} ...
3
votes
1answer
36 views

Verification of Proof that if f(x) is continuous and periodic then it is uniformly continuous on the reals.

Suppose f is defined on all reals. Then there is a positive p s.t. f(x+p)=f(x) for all x. This is my proof: Assume f is continuous on [0,p] then it is uniformly continuous on [-p,p]. Then for x,y ...
3
votes
2answers
55 views

Applications of the limit nth root of n.

I have the following question, and is that: Given that I've already proven that $$ \lim_{n \rightarrow \infty} \sqrt[n]{n}= 1 $$ Let $a_n = \sqrt[n]{n}$. I want to prove that $ a_n > a_{n+1} $, my ...
4
votes
3answers
73 views

Limit of $n(a^{1/n}-1)$ as $n \to \infty$

Show that $\lim(n(a^{1/n}-1)) = \ln(a)$ In the context of sequences, I'm not sure how to prove that this is the limit of the sequence. I was trying to convert the expression as follows: ...
0
votes
1answer
26 views

Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff $(a_n)_{n=m'}^{\infty}$ does.

Is my proof correct? Let $(a_n)_{n=m}^{\infty}$ be a sequence of real numbers. Let $c$ be real number. and let $m' \geq m$ be an integer. Show $(a_n)_{n=m}^{\infty}$ converges to $c$ iff ...
1
vote
2answers
48 views

Prove no member of $S = \left\{x \mid x \in \mathbb{Q^+}, x^2 < 2 \right\}$ can be an upper bound for $S$

Prove no member of $S = \left\{x \mid x \in \mathbb{Q^+}, x^2 < 2 \right\}$ can be an upper bound for $S$. So at first, I tried to prove that this statement was false; that a member of $S = ...
0
votes
2answers
69 views

Is it true that $|x^a - y^a| \leq |(x-y)^a|$ on $[0,1]$, where $a\le 1$?

It looks to me like for a function $f(x) = x^a$ on the domain $[0,1]$ where $a \leq 1$, and $x,y$ are points in the domain, $|x^a - y^a| \leq |(x-y)^a|$ I would like to use this in a proof and so if ...
0
votes
2answers
37 views

Prove that the limit of a series, containing 1/{powers-of-2}, is not rational

I have a series, $$x_n = \sum_{k=0}^n2^{-k^2-k}, \forall n \in N$$ I have to find it's limit and prove it is not in Q(it is not rational). I tried to write it $x_n=1+\frac{1}{2^1*2^1}+\frac{1}{2^4* ...
3
votes
2answers
51 views

Prove that $A$ is dense in $\bf R$

Let $A =\{\sqrt{m} - \sqrt{n }: \text{$n$ and $m$ are positive natural numbers}\}$. Prove that $A$ is dense in $\bf R$, I really can't find two natural numbers between $x$ and $y$. I tried ...
7
votes
0answers
88 views

Closed form of $\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$

Find the closed form of $$\sum_{n=1}^{\infty} \left(\frac{H_n}{n}\right)^4$$ I know the closed form for smaller powers like $2, 3$ exists, but I'm not sure if there is a closed form for this ...
0
votes
2answers
17 views

Is this the middle fourth cantor set?

Let $ D $ be the set of all $ x ∈ [0, 1] $ having a representation in the form $$ \sum_{i=1}^{\infty} {a_i}/{4^i} $$ where each $ a_i $ is either 0 or 3. Does this represent the middle fourth ...
1
vote
1answer
28 views

$\int_{0}^{1}(f'(x))^2-(f'(x))^3f(x)\; dx \geq 0$?

Is it true, that for all functions $f$ that $f(0)=f(1)=0$ : $$\int_{0}^{1}(f'(x))^2-(f'(x))^3f(x)\; dx \geq 0$$ I've tried to find counterexample, but I've not found.
0
votes
1answer
13 views

If $s_1 +s_2 \gt 1$ and $(t_1,t_2)$ be a convex combination of this with $(0.5,0.5)$ then show that $t_1t_2 \gt 0.25$

Let $(s_1,s_2)$ be such that $s_1 + s_2 \gt 1$. Let $(t_1,t_2)=((1-\epsilon )(0.5) + \epsilon s_1 , (1- \epsilon)(0.5) +\epsilon s_2)$, where $0< \epsilon \lt1$. I need to show that for $\epsilon ...
0
votes
1answer
42 views

Does uniform convergence imply $L^1$-convergence?

Suppose that $(X,M,\mu)$ is a measure space. Is it true that if $\,f_n \in L^1(X)\,$ and $\,f_n\,$ converges uniformly to $f$, then $\,f \in L^1(X)\,$ and $\,\int_X f_n\,d\mu\,$ converges to ...
0
votes
1answer
28 views

Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge.

Help please! Prove $\sum \sqrt{a_n b_n}$ converges if $\sum a_n$ and $\sum b_n$ converge. I can prove that $\sum a_n b_n$ converges but couldn't for $\sum \sqrt{a_n b_n}$. Thank you.
0
votes
0answers
28 views

Is the sequence convergent? [on hold]

Please give me a detailed explanation of this, consider me a layman in functional analysis! This question requires me to find whether the sequences are convergent in nature, using the following data. ...
-1
votes
2answers
30 views

Check for integrability of а function

Is the function $f(x,y)=\dfrac{\sin(x^2+y^2)}{x^2+y^2}$ integrable on $\mathbb{R^2}$ ? What is the general approach for investigating integrability?
1
vote
1answer
28 views

About the Fourier transform of the surface measure of the unit sphere

Let $d\sigma$ denote the surface measure on $\mathbb{S}^{n-1}$. To compute its Fourier transform $$ \hat{d\sigma}(\xi)=\int e^{-i x\cdot \xi}\, d\sigma(x), $$ a standard technique (cfr. Folland's ...
0
votes
1answer
17 views

Generated algebra and measurable functions

Define $\mathcal F := \{[n, n+1) : n \in \mathbb Z \} \subseteq \mathcal P(\mathbb R)$ and let $\Omega_{\mathcal F}$ be the $\sigma$-algebra generated by $\mathcal F$ on $\mathbb R$. Can someone ...
0
votes
1answer
29 views

Show function achieves an absolute minimum and an absolute maximum [on hold]

Suppose f:R-->R is continuous and periodic with period P>0. That is, f(x+P)=f(x) for all x ∈ R. Show that f achieves an absolute minimum and an absolute maximum.
0
votes
1answer
25 views

Switching the definition of epsilon-delta limit

What if the epsilon-delta definition of a limit reversed the wording for δ and ϵ: “for all δ>0, there exists an ϵ>0 such that, if 0<|x-a|<δ, then |f(x)-L|<ϵ.” Would this definition still ...
1
vote
0answers
38 views

How to show that : If $\int x^n f(x) dx = 0$ for all $n$ then $f(x) = 0$ for all $x$

How to show that : If $\int x^n f(x) dx = 0$ for all $n$ then $f(x) = 0$ for all $x$ My attempts: We can deduce that : $$\int \sum_i a_i x^i f(x) dx = 0 $$ for all $a_i$'s To the contrary if ...
0
votes
2answers
34 views

Is this an inequality law? with division

$a < b$ $c < d$ such that $a \ne b \ne c \ne d$ Will it be true that, $\frac{a}{c} < \frac{b}{d}$ For all positive $a, b, c, d$ Thanks!
0
votes
1answer
23 views

Is the composition of a measurable function with a monotone function measurable?

Assume that $\phi:\mathbb{R}\rightarrow\mathbb{R}$ is a strictly monotonically increasing function. Is it true that a real valued function $f:X\rightarrow\mathbb{R}$ is measurable on $(X,M)$ iff its ...
0
votes
2answers
30 views

subsequences and convergence for real analysis

Show there is a sequence $a_n$ such that for every real number x, there is a subsequence of $a_n$ converging to x. I have a hint that is to start with a bijection a:$\Bbb N$ $\to$ $\Bbb Q$.
0
votes
2answers
32 views

The sum of all vectors from two open sets gives an open set?

Let $A$ and $B$ be open subsets of $\Bbb R^k$. Then $A+B = \{\mathbf {x+y}: \mathbf{x}\in A \ and \ \mathbf{y} \in B\}$ is open in $\Bbb R^k$. I wanted to get an idea of what was going on first, so I ...
1
vote
1answer
26 views

Iterated Function System Question

Let $f_0(x) = \frac{1}{3}x^3$ and $f_1(x) = \frac{1}{3}x^3 + \frac{2}{3}$, where we consider both of these functions on $[0,1]$. These two maps are 'almost' Lipschitz contractions on $[0,1]$, but ...
0
votes
1answer
25 views

Are the following outer measures?

I have to check, if the following is an outer measures and if so, then determine the corresponding $\sigma$-algebra $\mathcal{A}(\mu^*)=\{A \subseteq X: A \text{ is } \mu^*\text{ - measurable }\}$. ...
0
votes
1answer
48 views

an example of a sequence $(u_n)_n$ taking its values in $[-1,+1]$ such that $(u_{n+1}-u_n)$ converge to zero but $(u_n)_n$ does not converge

Define a sequence $(u_n)_n$ by: $$u_n=\cos(\log n).$$ Then, it is easy to show that $(u_{n+1}-u_n)$ goes to zero at infinity. The question is how to prove that $(u_n)_n$ is a divergent sequence ...
0
votes
1answer
29 views

Derivative of $F : \mathbb R^{n+m} \rightarrow \mathbb R^{n+m}$

Question : Let f be continously differentiable function on an open set E $\subset \mathbb R^{n+m}$ into $\mathbb R^n$. Define a funtion F on E into $\mathbb R^{n+m}$ such that F(x,y) = ...
0
votes
1answer
21 views

continuity of a metric d

from Continuity of the Metric and Convergence Sequences, why $d^{-1}(V)$ is an open ball? to be an open ball, I think it contains elements of $X$, not $X^{2}$. why is it?
1
vote
1answer
36 views

Every sequence with $\lim x_n=c$, show that $f$ is continuous at $c$

Let $f:S\to \mathbb{R}$ be a function and $c \in S$, such that for every sequence ${x_n} \in S$ with $\lim x_n=c$, the sequence ${f(x_n)}$ converges. Show that $f$ is continuous at $c$.
0
votes
1answer
19 views

For$ f$ non-negative function, show that $f$ is meseurable if for some $p>0$ $f^p$ is measurable.

For$ f$ non-negative function ($f \geq 0$), show that $f$ is meseurable if for some $p>0$ the function $f^p$ is measurable.
1
vote
2answers
49 views

Does equality of antiderivatives imply equality almost everywhere?

If two Lebesgue integrable functions $\,f,g:[a,b]\to \mathbb R\,$ satisfy $$ \int_a^x f(s)\,ds= \int_a^xg(s)\,ds, $$ for every $\, x\in[a,b],\,$ then is is it true that $ f(x)=g(x)$ almost ...
0
votes
1answer
25 views

Find an integer $n$ such that: $(2^{p}-2)!+1=n((-√3+2)^{2^{p-2}}+(√3+2)^{2^{p-2}})$

Let $p$ be a prime number. My question is: Find an integer $n$ such that: $$(2^{p}-2)!+1=n((-√3+2)^{2^{p-2}}+(√3+2)^{2^{p-2}})$$ I can write $$(-√3+2)^{2^{p-2}}+(√3+2)^{2^{p-2}}=( ...
0
votes
2answers
30 views

Solving a simple integral by derivating w.r.t. to constants

In the following notes on the solution of the Wave equation by Separation of Variables, in Example 2 the following derivation is given \begin{align*} \int_0^1 x \sin(k\pi x) d x & = \int_0^1 ...
0
votes
1answer
17 views

Asymptotical behavior of $a_n$ with several properties.

Assume $a_n$ is infinite sequence. We know $a_n>0$, $\frac{a_n}{a_{n+1}}=1+\frac{p}{n}+c_n$ and $\sum \limits_{n=1}^\infty|c_n|$ converges. I want to prove that $a_n$ is asymptotically the same as ...
0
votes
1answer
26 views

Counterexample: if $f_n\rightarrow f$ in measure then $\frac{1}{f_n}\rightarrow \frac{1}{f}$ in measure

I was trying to find an example showing that this statement does not always hold. If a sequence of positive measurable function $f_n$ converges to $f$ in measure then $\frac{1}{f_n}$ converges to ...
1
vote
1answer
59 views

Is this an accurate limit proof for sine?

$\displaystyle \lim_{x\to 0} \frac x{1 + \sin^2(x)} = 0$ proof $$\left|\frac x{1 + \sin^2(x)}\right| < \epsilon$$ $$|x| < \delta$$ Let's require $|x| < 1$ so therefore, $$\sin^2(|x|) + 1 ...
0
votes
1answer
43 views

Limit proof for $1/x$ (as $x \to 1$)

Prove $\lim_{x\to 1} \frac{1}{x} = 1$ Using $\epsilon-\delta$ $|\frac{1}{x} - 1| < \epsilon$ for some $|x - 1| < \delta$ $|\frac{1}{x} - 1| = \frac{|1-x|}{|x|}$ Lets require $|x - 1| < ...
3
votes
1answer
53 views

Mean Value Theorem to Second Derivative

Let $f$ and $g$ be twice differentiable and continuous in $[a,b]$. Show that there exists a $c$ in $[a,b]$ such that: $$\frac{f(b) - [f(a) + f'(a)(b-a)]}{g(b)-[g(a)+g'(a)(b-a)]} = ...
1
vote
3answers
33 views

Limit Point Question

Question: Given a bounded sequence of real numbers, let $L$ denote the set of limit points of the sequence. Show that L is closed subset of$\mathbb R.$ Attempt: I was thinking of showing $L$ contains ...
3
votes
0answers
40 views

Inequality function

Let n $\in\mathbb{N} $ and $x_0,x_1,.....,x_n $ $x_i\in\mathbb{R}$ ,$x_i>0 $ so that $ x_0 + x_1 + .... x_n =1$. Prove that for all $a\in\mathbb{R}$ and $a>0$ the inequality is verified : $\ ...
4
votes
1answer
71 views

integral of a function

I wanted to find the integral of the function $f(x)$ from zero to one: $$f(x)=\begin{cases}2x\sin(1/x)-\cos(1/x) & : x\in(0,1]\\ 0 & :x=0\end{cases}$$ but I think whether its integral is not ...