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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2answers
29 views

Prove that $(a,b)$ is not compact based on the definition of compactness

I am trying to show that $(a,b)$ is not compact while $[a,b]$ is compact, purely based on definition of compactness. Here are examples, which I write it better: $[a,b]$ is covered by the intervals ...
0
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1answer
21 views

Limit of a sequence of a supremum.

Problem: Suppose that $f$ is continuous on $[a,b]$ and that $f(a)<f(b)$. Prove that there are numbers $c$ and $d$ with $a\leq c < d \leq b$ such that $f(c)=f(a)$ and $f(d)=f(b)$ and ...
1
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1answer
39 views

Is $[a,\infty )$ closed

In standard topology, of course $[a,\infty )$ is closed since its complement is open. But I don't know how to prove closeness of $[a,\infty )$ in Real Analysis using just the definition of closeness, ...
2
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0answers
29 views

Is it true?: Let $S$ be a subset of $\mathbb{R}$ wich is not sequentially compact. Then $S$ is not compact.

I just learnt the statement "Let $S$ be a sequentially compact subset of $\mathbb{R}$. Then $S$ is compact." with its relevant proof. My question is: Is it true to say "Let $S$ be a subset of ...
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1answer
17 views

If $∇f(a)\cdot y ≤ 0$ for every vector $y$, why does $\nabla f(a)$ have to be zero?

If $f$ is differentiable at every point in $B(a)$ and $f(x)≤f(a)$ for all $x$ in $B(a)$, prove that $∇f(a)=0$. I actually did some work and found out that $∇f(a)\cdot y ≤ 0$ for every vector $y$. ...
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0answers
16 views

Let $f$ be a scalar field such that $f ' (a ;-y)$ exsits [on hold]

Let $f$ be a scalar field where derivative of $f$ at point a with respect to vector $-y$ exists, $f '(a;-y)$ exists. Is it always true for any nonzero vector $y , f '(a;-y) = - f '(a;y)$?
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0answers
22 views

If $\mu(E)>0$ then $\exists E'\subseteq E$ such that $0<\mu(E')<\infty$.

Which measure $\mu$ have the property that for every measurable set $E$ with $\mu(E)>0$ there exist a measurable subset $E'\subseteq E$ such that $0<\mu(E')<\infty$? At first I thought every ...
1
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1answer
19 views

Connectedness, continuous functions, and the intermediate value theorem

I want to prove that for a continuous function mapping a connected space to ℝ such that f(p) never equals s, it follows that f(p) < s for all p or f(p) > s for all s. So here's what I know so ...
9
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1answer
47 views

Existence of a sequence of positive continuous functions on $\mathbb R$ which is unbounded precisely on $\mathbb Q$

Is it possible to find a sequence of positive continuous functions $g_n$ on the real numbers such that $( g_n(x) )$ is unbounded if and only if $x \in \mathbb{Q}$ ?
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0answers
43 views

Cantor set countable? [duplicate]

I know the Cantor set is uncountable, but I just came with an argument that shows it is countable. Obviously my argument is wrong, but I just don't know where is the mistake. Here it is. Let $C$ be ...
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2answers
37 views

If there is an $x$ such that $(2x_n)$ converges, does this imply that $(x_n)$ is also convergent?

I am toying around with the definition of convergence of sequences. Ans I asked myself the following question: Let $(x_n)$ be a real sequence such that there is an $x\in\mathbb{R}$ such that for all ...
0
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0answers
17 views

Constraint optimization with Calculus of Variations. How to handle positive function constraint?

the I am attempting to maximize the functional $F[f]$ with a constrain that $f$ has to be non-negative and some other integral constraints. More, specifically, \begin{align*} &\max F[f]\\ ...
0
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0answers
14 views

Application of Gauss divergence theorem

Put $\gamma(t) = (1+ \cos(t)) \begin{pmatrix} \cos(t) \\ \sin(t)\end{pmatrix}$, $t\in [0,2\pi]$. I would like to find the area enclosed by $\gamma$ by using the Gauss divergence theorem. Let $B$ be ...
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0answers
35 views

Proving $f=0$ if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ .

Let $f\in C^{\infty}[-1,1]$ and let $M$ be a constant such that $|f^{(j)}(x)|\le M$ $\forall j\in \Bbb{Z}_{+}$ and $x\in [-1,1]$. Prove that if $f({1\over k})=0$ $\forall k\in \Bbb{N}$ then $f=0$. I ...
-1
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1answer
23 views

Assume $f$ is differentiable at every point of $B(a)$ and $f(x)$ is less than or equal to $f(a)$

Over the scalar field, If $f$ is differentiable at every point in $B(a)$ and $f(x)$ is less than or equal to $f(a)$, prove why gradient of $f(a)$ is $0$. Just don't understand how to start with,
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3answers
54 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
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2answers
20 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
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0answers
34 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 ...
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0answers
21 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
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1answer
42 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 $\| \Lambda \|_2$ ...
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2answers
49 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
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1answer
30 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
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1answer
26 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
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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
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2answers
35 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
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3answers
46 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 ...
2
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1answer
79 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
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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
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1answer
18 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
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0answers
37 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
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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
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1answer
17 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 ...
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4answers
53 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
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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
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1answer
18 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 ...
3
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1answer
28 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 ...
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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.
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1answer
29 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 ...
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1answer
52 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
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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 = ...
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1answer
32 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
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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 ...
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0answers
41 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
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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
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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
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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 ...
4
votes
1answer
40 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
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3answers
87 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
22 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
70 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 = ...