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, and integration through the fundamental theorem of calculus.

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2
votes
2answers
21 views

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges.

Check whether or not $\sum_{n=1}^{\infty}{1\over n\sqrt[n]{n}}$ converges. I tried few things but it wouldn't work out. I would appreciate your help.
0
votes
1answer
11 views

Is distributional Derivative of $\delta^{(2)}(-x)=-\delta^{(2)}(x)$

Is distributional Derivative of $\delta^{(2)}(-x)=-\delta^{(2)}(x)$ ?? or $\delta^{(2)}(-x)=\delta^{(2)}(x)$ ?? I know that $\delta(-x)= \delta(x)$ and $\delta^{(1)}(-x)=-\delta^{(1)}(x)$. How ...
3
votes
0answers
24 views

Non-analytic smooth function

The Wikipedia page (http://en.wikipedia.org/wiki/Non-analytic_smooth_function) proves that $$f(x) = \begin{cases} \exp(-1/x), & \mbox{if }x>0 \\ 0, & \mbox{if }x\le0 \end{cases}$$ is a ...
0
votes
0answers
12 views

Arzela-Ascoli Theorem in $L^p[0,1]$

I understand the Arzela-Ascoli for $X$, compact metric space. So when $X=L^p[0,1]$, the theorem becomes the following? If $f_n\in C(L^p[0,1],L^p[0,1])$ that is uniformly bounded and equicontinuous, ...
2
votes
1answer
32 views

How to find the norm of $ \Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$ in $\ell^2$

Suppose that $(x_n)$ is a sequence in $\ell^2$, i.e. $\displaystyle \sum_{i=1}^{\infty} x_n^2 < \infty$. Define: $$\Lambda x = \sum_{n=1}^{\infty} \frac{x_n}{n\sqrt{6}}$$ Find $\| \Lambda \|_2$ ...
0
votes
1answer
37 views

Approximating solutions for the ODE $y'=\exp(y/x)$

I am currently trying to solve excercise 1-38 from Mathews and Walker. In this excercise I am asked to consider the differential equation: $$\frac{\mathrm{d}y}{\mathrm{d}x}=\exp(y/x)$$ for two ...
1
vote
1answer
26 views

Finding extrema of a continuous, univariate function.

Problem: Let $f:[0,1]\to\mathbb{R}$ be given by $f(x)=a(x-b)^2+c$, where $a,b,c$ are parameters. Find the minimum and maximum of $f$ depending on the values of $a,b,c$. I understand how to do this, ...
3
votes
1answer
25 views

Is $\|x\|^6 \sin^6 \|x\|^6$ harmonic?

Suppose the function $$ u(x)=\|x\|^6 \sin^6 \|x\|^6$$ for $x \in \mathbb{R}^d$, where $$\|x\| = \sqrt{x_1^2 + \ldots +x_d^2}.$$ How can I decide if the function $u$ is harmonic in the unit ball ...
-3
votes
0answers
40 views

Questions about Bolzano-Weierstrass Theorem [on hold]

Theorem 1.2. Let $\{x_n\}$ $(n=1,2,\ldots)$ be a sequence, and let $a$, $b$ be numbers such that $a \le x_n \le b$, for all positive integers $n$. There exists a point of accumulation $c$ of the ...
2
votes
2answers
34 views

Continuity of a Lebesgue indefinite integral over unbounded interval

We know that if $f : [a,b] \rightarrow \mathbb{R}$ is Lebesgue-integrable, then $$ F(x) = \int_{a}^{x} f(t) dt $$ is continuous. But if $f : \mathbb{R} \rightarrow \mathbb{R}$ is ...
3
votes
3answers
43 views

Prove this inequality.

Let $S=a_1+...+a_n<1$ where $a_i>0$. Prove that $1+S<(1+a_1)\cdot ... \cdot (1+a_n)<{1\over 1-S}$. I started with the right inequality but I am not sure it iss plausible (I did something ...
0
votes
1answer
48 views

Show that ${\{x_n}\}$ is convergent and monotone

Question: For $c>0$, consider the quadratic equation $$ x^2-x-c=0,\qquad x>0. $$ Define the sequence $\{x_n\}$ recersively by fixing $x_1>0$ and then, if $n$ is an index for which $x_n$ has ...
2
votes
1answer
13 views

Vector Cross Product and Expression for perpendicular distance between any two Vectors

If $B \ne C$, prove that the perpendicular distance from $A$ to the line through $B$ and $C$ is $$\dfrac {|| (A-B)\times(C-B)||}{||B-C||} $$ where $\times$ means the vector cross product. Attempt: ...
0
votes
1answer
17 views

Proof of floor function identity.

Let $f(x) = \lfloor x \rfloor$ and let $l$ be the greatest integer $\le x$ How do I prove $l + 1 > x$ I see that: $x \ge \lfloor x \rfloor = l$ No complete answers, just hints
1
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0answers
34 views

Proof of Geometric mean $\le$ Arithmetic mean

This is a problem from Spivak's Calculus. If $a_1, \dots, a_n \ge 0$, then the "arithmetic mean" $$ A_n= \frac{a_1+\cdots +a_n}{n}$$ and "geometric mean" $$ G_n= \sqrt[n]{a_1 \dots a_n}$$ ...
2
votes
1answer
20 views

Covering Lemma (Folland Lemma 3.15)

Lemma 3.15 from Folland's Real Analysis: Let $\mathcal{C}$ be a collection of open balls in $\mathbb{R}^n$, and let $U = \cup_{B \in \mathcal{C}}B$. If $c < m(U)$, there exist disjoint ...
3
votes
1answer
16 views

Other proof for existence of monotone subsequences

Is there any other proof of Bolzano-Weierstrass theorem (i.e.: Let ${\{x_n}\}$ be an arbitrary sequence of real numbers. Then ${\{x_n}\}$ has a monotone subsequence.), WITHOUT using concept of ...
1
vote
4answers
48 views

What are all values of $x$ in $\mathbb{R}$ that satisfy $4 < |x+2| + |x-1| < 5$?

I am having some problems getting started with this problem, as I never had to deal with an inequality that was between two values with absolute values. Any help is appreciated. The problem is find ...
1
vote
1answer
33 views

$\omega_f (x) = 0 $ iff f is continuous at $x$

Prove that $f$ is continuous at x iff $\omega_f(x) = 0 $ I need the help to prove the above statement. $\omega_f(x)$ is defined as: $\omega_{f}(x):= \lim _{\delta \rightarrow 0+} ...
2
votes
1answer
17 views

Limit to zero of the $p$-norm

I have the $p$-norm defined as $$\|x\|_p=\left(\sum_{k=1}^n|x_k|^p\right)^\frac{1}{p}.$$ I am trying to find the limit as $p\to0^+$ of $\|x\|_p$. I've seen it defined as $\{x_k:x_k\neq0\}$. Why is ...
2
votes
1answer
26 views

Proof of existence of “peaks” in The Sequential Compactness Theorem

According to Bolzano–Weierstrass theorem: How do we guarantee that EVERY sequence of real numbers has either infinitely or finitely many "peaks"? (Note that a subsequent is ordered in a way of ...
3
votes
1answer
30 views

Measure Theory, $\sigma$-algebra Folland Problem 23

I'm preparing for my exam. Can anyone help me in this matter, is confusing to me thank you very much.
4
votes
1answer
27 views

Requesting hints for showing that some function that is locally $L^p$ integrable is in $L^1(\mathbb{R})$.

Suppose $\int _a^b\vert f\vert^p<\infty$ for some $p\ge 1$ and for all $a,b\in \mathbb{R}$, and for some $a>p-1$ $$\int_{2\vert y-x\vert \le x}\vert f(y)\vert ^pdy\le \vert x\vert^{-a}$$ when ...
1
vote
1answer
50 views

Fundamental theorem of calculus, differentiable at the endpoints.

One version states: Let f be a continuous real-valued function defined on a closed interval $[a,b]$. Let f be the function defined for all x in $[a,b]$, by $F(x)=\int_{a}^xf(t)dt$. Then, F is ...
1
vote
1answer
15 views

Weak formulation of Poisson equation

I am learning about partial differential equations and I would like clarification on the weak formulation of the following 1D poission equation. Here is what I learned: $-u_{xx} = f(x)$ in $\Omega = ...
0
votes
1answer
29 views

If a subsequence of $(a_n)$ converges to a limit $L$, prove that $(a_n)$, which is non-decreasing, also converges to $L$

I know that if a sequence converges to $L$, its subsequences also converge to $L$. However, I'm not sure how to show the converse in this case, given $a_n$ is monotonic. If I show somehow that $a_n$ ...
1
vote
1answer
19 views

The measurability of $f(x) = \sum_{r_n \leq x} \frac{1}{2^n}$

Let $\mathbb{Q} \cap [0,1] = \{ r_1, r_2, \ldots \}$ be an enumeration of the rationals and let $f : [0,1] \rightarrow \mathbb{R}$ defined by $$ f(x) = \sum_{r_n \leq x} \dfrac{1}{2^n} $$ I need to ...
2
votes
0answers
40 views

Show a function is integrable on [0,1]

Let $f(p/q)=1/q$ if fraction $p/q$ is in lowest terms and $f(x)=0$ for irrational $x$ and when $x=0$. Show it is integrable on $[0,1]$. My idea: Construct a partition such that $U(p,f)-L(p,f) < ...
1
vote
1answer
14 views

Understanding Lipschitz domain

Here is the definition of Lipschitz domain given by Wikipedia. Let n ∈ N, and let Ω be an open subset of Rn. Let ∂Ω denote the boundary of Ω. Then Ω is said to have Lipschitz boundary, and is called ...
0
votes
1answer
25 views

Check that I didn't misuse the triangle inequality

This is a proof I am working on, and I think I've got it, but I wanted to make sure I didn't misuse the triangle inequality. I will point out what steps I think I may have made an implicit assumption. ...
0
votes
0answers
12 views

Can anyone help me prove derivative of scalar field using mean value theorem?

Assume that f′(x;y)=0 for every x in some n-ball B(a) and for every vector y. Use the mean value theorem to prove that f is constant on B(a). And if f′(x;y)=0 for a fixed vector y and for every x ...
3
votes
1answer
39 views

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$ [duplicate]

What is the period of $\sin 2\theta + \sin \frac{\theta}{2}$? The period of the first term is $\pi$ and that of the second is $4\pi$. Does that mean that the period of the whole is $4\pi$?
1
vote
3answers
84 views

What is valid and what is not in limits.

I have been looking for $$ \lim_\limits{x\to 0}{\left({\sin x\over x}\right)}^{1\over x^2}. $$ So I took $$ \lim_\limits{x\to 0}\left({\left({1+{\sin x-x\over x}}\right)}^{({x\over \sin ...
-2
votes
0answers
20 views

There is no scalar field such that $f '(a)>0$ for fixed $a$ and for every nonnegative vector $y$ [on hold]

I am trying to prove this. But can't think of how I should start. Anyone has some ideas? and why is there a scalar field $f'(a)>0$ for every $a$ and for fixed vector $y$ ? can anyone give me an ...
4
votes
4answers
68 views

Is this a valid way to show that the recursive sequence $x_n = x_{n-1} + \frac{1}{x_{n-1}^2}$ is unbounded?

I'm working through some analysis textbooks on my own, so I don't want the full answer. I'm only looking for a hint on this problem. Rosenlicht's Introduction to Analysis asks me to prove that $x_n = ...
7
votes
2answers
174 views

Am I wrong for wanting to give up studying analysis?

I am writing this because I'm having a hard time studying for analysis. This is my second time taking a real-analysis course and from the beginning, somewhere deep down I thought that I might be ...
1
vote
1answer
34 views

Prove that $\left| f'(x)\right| \leq \sqrt{2AC}$ using integration

Suppose that $f(x)$ is a $C^2$ function on $\mathbb{R}$ such that $\left| f(x) \right| \leq A$ and $\left| f''(x) \right| \leq C $ for $x \in \mathbb{R}$. Prove that $\left| f'(x)\right| \leq ...
0
votes
0answers
15 views

Redundancy in the Laplace transform and Mellin's inverse formula

As I understand it, Mellin's inverse formula relates a sufficiently 'nice' function $f$ and its Laplace transform $F$ as follows: $$f(t)=\frac1{2\pi i}\lim_{T\to\infty}\int_{-T}^{T}e^{i\omega ...
2
votes
2answers
19 views

Fixed Point Iteration Method

Can anyone explain or prove the Fixed Point Iteration method? I know the conditions of fixed point existence. A fixed point is a point of a function ${f}$ on a continuous interval ${(a,b)}$ which ...
1
vote
1answer
40 views

Applying the Cauchy Swartz Inequality

Let $A = (a_{ij})$ be an $n \times n$ real matrix, $I = (\delta_{ij})$ the $n \times n$ identity matrix, $b \in \mathbb{R}^n$. Suppose that $$\|A - I\|_2 = \left(\sum_{i=1}^n \sum_{j=1}^n (a_{ij} - ...
2
votes
3answers
26 views

Prove that $\sqrt x$ is Lipschitz on $[1, \infty)$

Prove that $\sqrt x$ is Lipschitz on $[1, \infty)$ I want to show that $|f(x) - f(y)| \leq L |x - y|$ So $|\sqrt x - \sqrt y| = \frac{|x - y|}{\sqrt x + \sqrt y} \leq \frac{1}{2}|x - y|$. I can ...
0
votes
2answers
42 views

Compute $\lim_{|h|\rightarrow \infty} \int_{\mathbb{R}^n} |f(y+h)+f(y)|^p dy$

Suppose $f\in L^p(\mathbb{R}^n), 0<p<\infty$, and compute $\displaystyle \text{lim}_{|h|\rightarrow \infty} \int_{\mathbb{R}^n} |f(y+h)+f(y)|^p dy$ I have no idea from where to start since ...
1
vote
0answers
13 views

Cavalieri's principle and integrals

Let $f$ be a continuous function an $[a,b]$. Let $P\subset R^2$ be the figure under the graph of $f$ and $D \subset R^3$ a solid figure obtained by rotating a plane curve around $x$-axis. Using ...
0
votes
0answers
16 views

Iteratively solve this equation

I am supposed to solve $1 = \left( \frac{\mu}{f} \right)^{\frac{3}{2}} \left( 1+ \frac{ \pi^2}{8} \left( \frac{kT}{\mu} \right)^2 \right)$ iteratively for $\mu$ and am supposed to get $$\mu = f ...
1
vote
0answers
22 views

Lebesgue-Stieltjes Measure associated to $F$.

I would like some help here, please. First is confusing to me the definition of: Lebesgue-Stieltjes Measure associated to $F$. I'm reading Folland-Real Analysis, page 35, second paragraph. I do ...
0
votes
0answers
10 views

A question about n-dimensional Lesbegue measure

This is the p.70 of Folland Real analysis. It says that the completion of $m^n$ on $\mathcal{B}_{\mathbb{R}^n}$ is equal to the completion of $m^n$ on $\mathcal{L}^n$. But Why do they agree? Could ...
2
votes
4answers
60 views

Find $y^{(n)}(0)$ for every $n$.

Let $y(x)$ fulfill $y''-xy=0$. Furthermore: $y(0)=0,y'(0)=1$. Find $y^{(n)}(0)$ for every $n$. I tried different forms of recurrence relations but I couldn't do much with it without it becoming a ...
3
votes
1answer
37 views

Weak convergence - $f_n$ “goes up the spout”

Fix $1 < p < \infty$. Given $f \in L^p(\mathbb{R})$ define $f_n(x) = n^{1/p}f(nx)$ for $n = 1, 2, \dots$. Prove that $f_n$ converges weakly to $0$ in $L^p$. I'm really confised about this ...
1
vote
1answer
26 views

Confused about this definition of limit superior.

The definition I am given is as follows: Let $(x_n)$ be a real valued sequence. For each positive integer $n$, let $s_n:=\sup\{x_m:m\geq n\}$. If $(s_n)$ converges, we denote its limit by ...
0
votes
2answers
35 views

is this correct that if $\frac{\partial f}{\partial y}=0$ then $f$ is independent from $y$?

Suppose that $A=\{(x,y) \in \Bbb R^2 : x> 0 $ or $ y=0 \}$ and $f:A\to \Bbb R$ is an arbitary function. Prove that If $\frac{\partial f}{\partial x}=0$ then $f$ is independent from $x$ If ...