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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-9
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0answers
32 views

Real Analysis-Find the limit of this series [on hold]

Problem # 3 enter image description here Find the limit of this series.
2
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2answers
26 views

Question about proof: Uniform cauchy $\Rightarrow$ Uniform convergence

I have one quick question regarding the proof of a theorem contained in here : https://www.math.ucdavis.edu/~hunter/m125a/intro_analysis_ch5.pdf Theorem 5.13. A sequence $(f_n)$ of functions $f_n ...
0
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1answer
16 views

Growth rate of large sets

Suppose that ${a_k}$ is a real valued increasing sequence such that $$ \sum_{k=1}^{\infty} \frac{1}{a_k} = +\infty ,$$ i.e. $\{a_1,a_2,\ldots\}$ is a large set. If $\lim a_{k+1} - a_{k} = \infty$, ...
2
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2answers
48 views

Proof of $\lim_{h\to 0}\frac{f(a+ h)-f(a)}{h}=\ell$ when $\lim_{x\to a}f'(x)=\ell$

Suppose $f$ derivable on $\mathbb R$ and that $\lim_{x\to a}f'(x)=\ell$. Show that $$\lim_{h\to 0}\frac{f(x+h)-f(x)}{h}=\ell.$$ The proof of my course goes like this: By mean value theorem, there is ...
1
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1answer
23 views

Several questions about Riesz–Markov–Kakutani representation theorem

This is a list of questions about Riesz–Markov–Kakutani representation theorem . 1)If $f\in L^1(\mu)$, is it true that $\phi(f)=\int_Xfd\mu$, where $\mu$ is given by the theorem? I am quite sure it ...
3
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3answers
30 views

Convergent sequence of irrational numbers that has a rational limit.

Is it possible to have a convergent sequence whose terms are all irrational but whose limit is rational?
0
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0answers
24 views

Can I use the Squeeze theorem with sequences of functions?

For example if I know that $f_n(x)\leq g_n(x) \leq h_n(x)$ for all $x$, then can I say that $ \lim_{n \rightarrow \infty} f_n(x)\leq \lim_{n \rightarrow \infty} g_n(x) \leq \lim_{n \rightarrow \infty} ...
1
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1answer
27 views

Maximal interval of solutions existence: $x'(t)=-x(t)+\sin x(t)+t^3$

$x'(t)=-x(t)+ \sin x(t)+t^3$ in $\mathbb{R}$ I consider the function: $$ f(t,x)=-x+\sin x + t^3 $$ $$\frac{\partial f}{\partial x}=\cos x-1$$ I see that: $$\left| \frac{\partial f}{\partial x} ...
0
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0answers
10 views

How can I find a mapping $G(x,y)$ [on hold]

How can I find a mapping $G:X\times X \rightarrow X $ (where $X\subset R$) such that $G(x,y)$ is non-increasing in arguments $x$ and $y$ and $G(x,x)=0$ for all $x\in X.$ Thanks
5
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2answers
34 views

Deducing the series expansion of $\arctan(x^2)$ via the series expansion of $\arctan(x)$ at $x=0$

Comparing the series expansion of $\arctan(x^2)$ and $\arctan(x)$ at $x=0$ it looks like one can take the result from $\arctan(x)$ and replace each $x$ with $x^2$ to deduce the series expansion of ...
1
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0answers
10 views

About Convergence of a series [on hold]

Is this series convergent? $$\sum_{n=0}^{N-1}\frac{c_{n}^{N}}{c_{N}^{N}n^2}$$ where $c_{n}^{N}$ is coefficient $x^{n}$ in chebyshev polynomial $T_{N}(x)$, i.e. ...
0
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0answers
16 views

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable for $p>0$

Show $A=\{x\in \Bbb{R}^n|\sum_{j=1}^{n}|x_j|^p\le 1\}$ is Jordan measurable if $p>0$. I did show it is a bounded set because if there exists $x^{(N)}\subset A $ such that $||x^{(N)}||\to \infty $ ...
0
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1answer
15 views

$X$ sequentally Compact implies that $X$ is complete

I was reading through Roydens book and there is one part that I don't understand. Here is the proof. Suppose $X$ is sequentially compact metric space, then $$X \text{ is sequentally compact}:= ...
0
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0answers
16 views

Question About Representation of Brownian Motion

In Stochastic Calculus with Financial Applications by J. Michael Steele he makes a specific representation of a Brownian Motion using wavelets. At one point he calculates the covariance, and he uses ...
0
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3answers
45 views

For $\int f < \infty$, the measure of the set of points where $f=\infty$ is zero.

I fear this question was already discussed here, but I was not able to find it. Please remove if it is a duplicate. Prove: For a function $f\geq 0$, if $\int f < \infty$, then the measure of ...
0
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1answer
8 views

Finding First Integrals in the case $2xy u_x - (x^2+y^2) u_y =0$

Good day, As described in the title, I want to find two First Integrals (FI) to the PDE $$2xy u_x - (x^2+y^2) u_y =0$$ Of course, $u$ is a FI and the solution of the PDE ist $u(x,y)=u_0$. But I want ...
0
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2answers
34 views

Unit closed ball

I had the following question: Maximum $\{x +y : (x,y) \in closure [B(0,1)] \}$. Here $B(0,1)=\{(x,y) \in \mathbb{R}^2 : x^2+y^2 \leq 1\}$ I can't proceed in this as to how to maximize this value. ...
1
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1answer
35 views

Determine supremum and infimum of $\{x \in \Bbb R\,:\, x < 3/x\}$

Set $S := \{x \in \Bbb R\,:\, x < 3/x\}$. (a) Determine whether $\sup S$ exists, and determine its value if it exists. Justify your answer. (b) Determine whether $\inf S$ exists, and determine ...
2
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3answers
58 views

Calculate $\int_0^1 \ \int_0^1 \ x \sin \lvert x^2-y^2 \lvert \; dx \; dy$

$$\int_0^1 \ \int_0^1 \ x \ \sin \lvert x^2-y^2 \lvert dx \ dy $$ $$\int_0^1 \frac{1}{2} \Big[ \sin \lvert x^2-y^2 \lvert \Big]_0^1 \ dy= \int_0^1 \frac{1}{2} \Big( \sin \lvert 1-y^2 \lvert - ...
4
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5answers
53 views

About inner products, norms and metrics

Do these three kinds of vector spaces, those with an inner-product, those with a norm and those with a metric, are the same sets of vector spaces? At least for finite dimensional vector spaces all of ...
2
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1answer
25 views

$\sigma$-algebra produced by a subclass of a class.

im studying the book 'probability & measure' by Patrick Billingsley. in chapter 2 there's an exercise 2.9 say's: show that: If $B\in\sigma(A)$, then there exists a countable subclass $A_B$ of $A$ ...
0
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1answer
33 views

How to establish convergence and find limit of the sequence$(n+1)^{1/\ln(n+1)}$ [on hold]

How to establish convergence and find limit of the sequence $(n+1)^{1/\ln(n+1)}$. I know its a stupid question but its kinda urgent so please help me out! Edit 1: It is urgent because I have to ...
1
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3answers
61 views

Show that f is measurable.

Let $a > 0, b \geq 0$ and the function $f: \mathbb{R} \to \mathbb{R}$ $$f(x) = \left\{\begin{matrix} 1, & |x| \leq a \\ b & |x| > a \end{matrix}\right.$$ show that it is measurable. ...
3
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1answer
43 views

Is $g(x)$ uniformly continuous?

$f:\mathbb R\to[0,\infty )$ is continuous such that $g(x)={(f(x))}^2$ is uniformly continuous. Then prove that $f(x)$ is uniformly continuous. I've tried to use the following method: $$ ...
0
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0answers
40 views

Prove that $f'(c)= \frac{2}{2+3(f(c))^2}$ for some $c$

Problem: $f: [0, 1] \to \mathbb{R}$ is continuous on $[0, 1]$ and differentiable on $(0, 1)$. ALso, $f(0)=1$ and $(f(1))^3+2f(1)-5=0$. Prove that there exists a $c \in (0, 1)$ such that $f'(c)= ...
3
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2answers
49 views

Construct $f: X\to Y$ such that $f(p)=p$

Let , $X=[-1,1]\times [-1,1]$ and $Y=\{0\}\times \left[-\frac{1}{2},\frac{1}{2}\right]$. Construct an example of a continuous map $f:X\to Y$ such that $f(p)=p$ for each $p\in Y$. I construct a ...
0
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3answers
24 views

How do I decide which convergence criterium to use and what to do if they don't work?

Take $$ \sum_{n=1}^\infty (-1)^nn^{1/n} $$ I just blindly tried (Ratio test) $$ \limsup_{k\to\infty} \left|\frac{a_{k+1}}{a_k}\right| $$ and (Root test) $$ \limsup_{k\to\infty} \sqrt[k]{|a_k|} $$ ...
1
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1answer
12 views

Check if this is the example of $x$ as a limit point of $C$ but ($x_t$) does not converge to $x$.

Let ($x_t$) be a sequence in a metric space, and let $C$ be the range of ($x_t$). I want to give an example in which $x$ is a limit point of $C$ but ($x_t$) does not converge to $x$. Here's my ...
2
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0answers
15 views

Boundary of a $k-$Cell Definition

In the book I am reading, the Boundary of a $(k+1)-$cell $\varphi$ is defined to be the $k-$chain $$\partial\varphi=\sum_{j=1}^{k+1}(-1)^{j+1}(\varphi\circ \iota^{j,1}-\varphi\circ \iota^{j,0}) $$ ...
1
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1answer
39 views

prove that if $a>1$ , then the limit of $\frac{a^n}{n}$ is $\infty$ as n goes to $\infty$

prove that if $a>1$ , then the limit of $\frac{a^n}{n}$ is $\infty$ as n goes to $\infty$ I was trying to use the binomial theorem to replace $a$ with $1+k$, but then that $n$ on the denominator ...
1
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1answer
33 views

prove the following limit without evaluating the integrals explicitly

$\lim_{n\to \infty}\int_1^2 \ln^n(x) \, dx=0$ $\lim_{n\to \infty}\int_2^3 \ln^n(x) \, dx=\infty$ I know for #2 I need a lower estimate for the integral that is large to show that the limit is ...
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0answers
11 views

Setting Lineard Equation ODE into standard form, proving existence of unique solutions

I'm working on proving existence and uniqueness of a local solution to ODE's of the Lienard variety $$y'' + f(y)\,y' + g(y) = 0$$ I'm trying to put this into standard $y' = f(t,y)$ system So I see ...
1
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4answers
52 views

Is the ''right limit'' function always right continuous?

Let $f$ be a bounded function on $[0,1]$. Assume that for any $x\in[0,1)$, $f(x+)$ exists. Define $g(x)=f(x+)$, $x\in [0,1)$, and $g(1)=f(1)$. Is $g(x)$ right continuous? Prove it or give me a ...
2
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1answer
18 views

Turing criteria for Sel'kvo glycolysis model

I have the Sel'kov reaction diffusion model for glycolysis as follows: \begin{eqnarray} u_t=D_uu_{xx}-u+av+u^2v\\ v_t=D_vv_{xx}+b-av-u^2v \end{eqnarray} How can I obtain the values for $D_u$ and ...
0
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0answers
21 views

Find the integrator satisfying an identity for all continuous $f$

Let $\alpha: [0,1] \rightarrow \mathbb{R}$ be a monotone increasing function and $f \in \mathcal{R}(\alpha)$. Find $\alpha$ such that $$\int_0^1 f(x)d\alpha=\int_{1/2}^1 f(x)dx+f(1/2)$$ I've ...
0
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2answers
41 views

Show analytically that $te^{-t}$ is not decreasing monotonically.

How does one show analytically that $te^{-t}$ is not decreasing monotonically on $(0, \infty)$? One can consider numbers in the interval $(0, 1]$ and show a counterexample to monotonicity, but ...
2
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0answers
29 views

A $k+1$ Manifold whose boundary is the solution set to the equation $f(\vec{x})=\vec{0}$

Let $f: \mathbb{R}^{n+k} \to \mathbb{R}^n$ be of class $C^r$. Let $f_1, ..., f_n$ be components of $f$. Define $$M=\{\vec{x} | f(\vec{x})=\vec{0}\},$$ $$N=\{\vec{x} | f_1(\vec{x})=0, ..., ...
1
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1answer
13 views

Independence of two non-negative integer valued random variables

Let $X,Y$ be two non-negative integer valued random variables defined on a probability space $(\Omega,\cal F, \Bbb P)$. The question is, If $\Bbb P\{X=i,Y=j\}=\Bbb P\{X=i\}P\{Y=j\}$ for every ...
0
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0answers
16 views

Give an example of limits that misbehave under conjugation of function

My quest: Find real valued functions $f(x)$ and $g(x)$ such that $f \rightarrow b$ as $x\rightarrow a$ and $g\rightarrow c$ as $x\rightarrow b$ but $g(f(x)) \nrightarrow c$ as $x\rightarrow a$ I ...
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0answers
34 views

The closure of the closed ball is a closed [on hold]

In how many ways you can show that $\overline{\overline{B}(x,r)}=\overline{B}(x,r)$ where $\overline{B}(x,r)=\lbrace y \in \mathbb{R}^n : d_e(x,y) \leq r \rbrace$, and $d_e$ is the euclidean metric ? ...
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1answer
20 views

Stable eigenspace of $x'=Ax$

Given the system $\bf{x'}=Ax$, where $\bf{A}$$=\begin{bmatrix} -2 &0 &0 \\ 2& 1 & 0\\ 0 &0 &1 \end{bmatrix}$, the solution is $x(t) = \begin{bmatrix} e^{-2t} & 0 ...
1
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0answers
19 views

Convergence of Fourier sine and cosine series

Discuss whether or not it is possible to have a Fourier series $$a_0+\sum_{k=1}^\infty[a_k\cos(kx)+b_k\sin(kx)]$$ converge for all $x$ without either $$a_0+\sum_{k=1}^\infty a_k\cos(kx) ...
0
votes
1answer
24 views

Strictly monotone functions and continuity

Let $f : X \to I$ be a strictly monotone surjective function mapping $X \subseteq \mathbb{R}$ to an interval $I \subseteq \mathbb{R}$. Then is $f$ necessarily continuous? Without loss of ...
0
votes
1answer
22 views

Real Analysis, Folland 3.4.26, Differentiation on Euclidean Space

Background Information - A Borel measure $\nu$ on $\mathbb{R}^n$ will be called regular if i.) $\nu(K) < \infty$ for every compact $K$ ii.) $\nu (E) = \inf\{\nu(U): E\subset U, U \ ...
0
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1answer
45 views

Is the intersection of all intervals $\left( 0, \frac{1}{i} \right)$, where $i$ is in $\{1,2,3…\}$, equal to the empty set?

So, does $\big(0,\frac{1}{1}\big)\bigcap\big(0,\frac{1}{2}\big)\bigcap\big(0,\frac{1}{3}\big)\bigcap\dots = \emptyset$ ? This has bugged me for the last couple of hours. The question is whether ...
4
votes
0answers
49 views

Is the sum of infinitely many open sets open?

Let $X$ be a locally convex space (or, in particular, a normed space). Let $(O_n)_{n=1}^\infty$ be an infinite sequence of non-empty open sets in $X$ such that the sum $\displaystyle\sum_{n=1}^\infty ...
4
votes
4answers
290 views

quick question on an example of the derivative as a linear map.

After reading many answers on the subject I feel like I am close to finally understanding why the derivative is a linear map. I think that if someone helps me understand the following example I might ...
1
vote
1answer
44 views

$x'=\cos^5(x) +1$ has unique solution defined for all $t\in \mathbb{R}$

I would appreciate if someone could please give me a hint on how to do this problem. Or where to see some examples. Unfortunately, the sources that I have do not seem to actually explain it and show ...
0
votes
1answer
21 views

Proving That $\alpha^{\prime}(t)$ is orthogonal to $N(\gamma(t))$

Let $\Phi$ be a surface in $\mathbb{R}^3$ with parameter domain $K\subset\mathbb{R}^2$ and let $\gamma:[a,b]\to K$ be a $\mathcal{C}^1$-curve. Also let $\alpha=\Phi\circ\gamma$. Prove that ...
1
vote
3answers
24 views

Uniform convergence with two limits

I'm doing a question investigating uniform convergence of a function and I need something cleared up if possible. $f_n(x) = \frac{x^n}{1+x^n}$ on the interval $[0,1]$. Now, pointwise, this turns ...