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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The relationship between function space embeddings and their respective inequalities

Let $L^{p,\infty}$ be the weak $L^p$ space consisting of measurable functions $f$ satisfying \begin{equation*} ||f||_{p,\infty}:=\sup_{\rho}\rho\lambda (|f|>\rho)^{\frac{1}{p}}<\infty . ...
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3 views

Approximation by $\mbox{Im }(t-z)^{-1}$ with $\mbox{Im } z > \epsilon$

It is a standard fact of harmonic analysis that the span of the functions $$g_z(t) = \mbox{Im } (t-z)^{-1},$$ ranging over all $z \in \mathbb{C}$ with $\mbox{Im } z > 0$, is dense in ...
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24 views

If $f'(ax) = \frac{b}{a}$ then $\frac{\textrm{d} x}{\textrm{d} a} > 0$ where$f, f' > 0$ and $f'' < 0$?

I'd appreciate any help I could get for this question: Define $f:[0, +\infty) \rightarrow R^+$ with $f(0) = 0$ and $f(\infty) = \infty$, and such that $f' > 0$ and $f'' < 0$. I have the ...
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0answers
15 views

$e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$

I have to show that $e\cdot(m(m-1)+1)\cdot k\cdot ( 1-\frac{1}{k})^m\leq 1$ holds for all positive integers k and m whenever $m>4\cdot k\cdot log(k)$. I replaced $(1-\frac{1}{k})^m$ with ...
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0answers
8 views

Show measures are aboslutely continuous based on relationship between integrals of certain functions

So I have the following question. Suppose $\mu$ is a regular Borel measure on $[0,1]$ and $\{f_n\}\subseteq L^1[0,1]$ s.t. $|f_n(x)|\leq |f(x)| \ [m]$ a.e. and $f\in L^1[0,1].$ Given ...
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1answer
40 views

An example of discontinuous function on $\mathbb{R}$ [duplicate]

Is there any example of a function which is discontinuous on $\mathbb{R}$
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7 views

Least expensive way to “walk” with a convex potential

Let $V(x)$ be a non-decreasing, convex function on $\mathbb{R}^+$ such that $V(0)=0$, $V(1)=1$. Let $A_{L,k}$ be the space whose elements are sequences of positive integers $r= (r_i)_{i=1}^{N}$ such ...
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9 views

A pde that cannot solves by Lax-Milgram theorem

Consider the following pde: $-u''(x)+au'(x)+bu(x)=f(x) \qquad\text{in}\; (0,1)\\$ $u'(0)=\alpha\\$ $u'(1)+u(1)=\beta$ How could I prove that it has a nontrivial solution? The bilinear form ...
3
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1answer
24 views

Manipulating the maximum function, metric spaces.

I am trying to show that the supremum metric, $d_{\infty}$, is indeed a metric on $\mathbb R^2$. I have shown that the first two properties of a metric space hold, but am having trouble showing the ...
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24 views

Proof of additivity of domain for definite integrals

I would like to prove the following theorem: Theorem If $c \in (a,b)$ and $f$ is integrable on $[a,c]$ and $[c,b]$, then $f$ is integrable on $[a,b]$ and $$\int_{a}^{b}f = \int_{a}^{c}f + ...
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3answers
36 views

The derivative function is not continuous

(Sorry about the bad title, couldn't think of a way to word it concisely.) Let $C[0, 1]$ be the metric space whose points are all continuous functions from $[0, 1] \rightarrow \mathbb{R}$ with the ...
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34 views

Pugh real mathematical analysis - chapter 1 - exercise 14 [on hold]

I absolutely have no idea for solving the problem... any help would be appreciated!
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1answer
30 views

Show that $\lim_{t\to \infty}u(x,t)=\frac{A+B}{2}$, for each $x\in\Bbb R$.

Let $u(x,t)$ be a $C^2$ bounded solution of $$u_t(x,t)-u_{xx}(x,t)=0,x\in \Bbb R, u(x,0)=f(x)$$ where $f\in C(\Bbb R)$ satisfies: $\lim_{x\to+\infty}f(x)=A,\lim_{x\to-\infty}f(x)=B$. Show that ...
5
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82 views

Cauchy-Schwarz Inequality without using $\langle a x,y\rangle=a\langle x,y\rangle$

Let $V$ be a vector space and define a function $\langle .,.\rangle:V\times V\to\mathbb{C}$ such that $$\begin{align} & \langle x,y\rangle=\overline{\langle y,x\rangle }\,\,\,\forall x,y\in V\\ ...
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23 views

Let $s$ be a simple function st. $s\le g+h$, show there exists simple functions $u\le g$, $v\le h$ st. $s=u+v$

Let $g,h$ be two functions and let $s$ be any simple function such that $s \le g+h$. Show there exists simple functions $u\le g$ and $v\le h$ such that $s=u+v$. I have been thinking about this ...
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0answers
16 views

Additional assumptions on function to ensure uniform convergence

Given a sequence $u=(u_n)_{n\geq1}$ converging to $1$, I would like to prove uniform convergence of the sequence of functions $f_n$ defined by $f_n(x)=f(u_n x)$ for $x\in\mathbb{R}_+$ to the function ...
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1answer
32 views

If I have a function that's continuous and it's limits at $\pm \infty$ are $\pm \infty$ is it surjective?

I was trying out some problems where I needed to prove that a function was surjective, and I thought I could do this, is this true? Intuitively, it seems so. If I have a function that's continuous ...
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3answers
41 views

An exercise on compactness on $L^1$

If $g$ is a nonnegative measurable function on $[0,1]$, and let $K=\{f\in L^1([0,1]) :|f|\leq g \,\,\,\, a.e.\}$, prove the following: (a)$K$ is closed, (b)If $K$ is compact then $g\in L^1$. Part ...
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4answers
58 views

For any closed subset of $\mathbb{R}$ there is a sequence in $\mathbb{R}$ whose sequential limits is equal to the that subset

Question: Let $A$ be a closed subset in $\mathbb{R}$. Prove that there exists a sequence $x_n$ in $\mathbb{R}$ whose set of subsequential limits is exactly equal to $A$. My approach: I think this ...
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1answer
77 views

Help with a proof regarding simple functions.

The question is If $f>g≥0$, then there exists non-negative measurable simple functions $f_k↗f$ s.t. $f_k≥g$ for all $k$. My attempt. Using a theorem in my text book For every ...
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2answers
90 views

Condition to guarantee $f=0$ on $[a,b]$

I have been stuck for several days on this old Analysis problem (I am doing some study on my own). I have tried several things (which I'll indicate below), but I cannot seem to figure it out. Here ...
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2answers
53 views

Is the Commutative Property of Addition on the Reals a Postulate or Proven?

My Calculus book from back in the day (Calculus Second Edition Michael Spivak) starts out by stating 12 basic properties of numbers which he labels P1-P12. He states: "Most of this chapter has ...
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0answers
23 views

Maximal cake-cutting

Alice and George divide a cake between them. The cake is a 1-dimensional interval and both players value the entire cake as 1. The valuations of the players are represented by non-atomic measures on ...
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1answer
44 views

The Real Numbers and Real Analysis

Prove that $\sqrt{1+4x}<2x+1$ for all $x\in(0,\infty)$ using the Intermediate Value Theorem of Derivatives.
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1answer
43 views

In which sense does Cauchy-Riemann equations link complex- and real analysis?

On page 12 of Stein, Shakarchi textbook 'Complex analysis', the authors state that the Cauchy-Riemann equations link complex and real analysis. I have completed courses on real and complex analysis, ...
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3answers
20 views

Fixpoints and continuity

I don't understand why this is true: If $f:[0,1]\rightarrow[0,2]$ is a continuous function then exists $x \in [0,1]$ such that $f(x)=2x$ I don't understand why such a point exist. Why is there not ...
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37 views

Need help solving min, max, inf and sup of sequence!

We've been given the sequence $x_n=(-1)^n \cdot \frac{\sqrt{n}}{n+1}+\sin \frac{n \pi}{2}$. I have to find the min, max, inf and sup (if they exist), and also find the points of accumulation. Any ...
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21 views

Hausdorff and surface measure

I know that Lebesgue measure in $\mathbb{R}^n$ coincides with Hausdorff measure $\mathcal{H}^n$. But I'd like to see the proof that also surface measure on manifold $M$ coincides with the Hausdorff ...
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0answers
21 views

Limit of the “oscillation” of a function

Let $f:[a,\infty) \rightarrow \mathbb R$ be a bounded function, and for each $t\geq a$ the functions $M_t = \sup f|_{I}$, $m_t = \inf f|_{I}$ where $I=[t,\infty)$. Define the function $\omega_t = M_t ...
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2answers
69 views

How to show $\int_{[0, +\infty)} \frac{2}{1+x^2} dx$ Lebesgue integrable?

Definition of Lebesgue integral of simple function: We say that a simple function $\psi$ is Lebesgue integrable if the set $\{\psi \ne 0\}$ has finite measure. In this case, we may write the ...
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1answer
30 views

How to know which notion of convergence to use when proving density of a subspace

My question might be a little vague, but is there a way to know which type of convergence (i.e pointwise, uniform) to use when proving that a subspace is dense in a certain space. For example if we ...
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4answers
83 views

On extended real line, is $(-\infty,+\infty)$ still a closed set?

On real line $(-\infty,+\infty)$ is open as well as closed. On extended real line $[-\infty,+\infty]$, is $(-\infty,+\infty)$ still a closed set? Thank you.
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1answer
62 views

Closed form of series involving Fibonacci numbers

Let $F_n$ denote the $n$-th Fibonacci number and $\phi$ be the golden ratio, that $\phi = \frac{1+\sqrt{5}}{2}$. Find a closed form for the sum: $$\sum_{n=0}^{\infty} \frac{1}{(5\phi)^n(n+2)} ...
6
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1answer
121 views

Finding a point with $f(x)=f'(x)$

Let $f :[0,1] \rightarrow \mathbb{R}$ be a fixed continuous function that is differentiable on $(0,1)$ and such that $f(0)=f(1)=0$. Does there exist a $x_0 \in (0,1)$ such that ...
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1answer
24 views

Another question on finiding special kind of power series [on hold]

Let $\sum a_nx^n$ be a real power series with finite positive radius of convergence $R$ ; then does there exist a non-constant real sequence $\{b_n\}$ such that $\sum b_nx^n$ is convergent for at ...
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0answers
41 views

Theorem 12.13 of Apostol's Mathematical Analysis (2nd Ed)

I had a little difficulty understanding Theorem 12.13 of Apostol's Mathematical Analysis, and was wondering if someone who had read this book could give me some pointers. Specifically, on page 360, ...
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1answer
38 views

Sequence equivalence [on hold]

We call two sequences are equivalent if $\lim_{n\rightarrow\infty}u_n/v_n=1$. (1) Using $e=\sum_{k} 1/k!$, find a simple, nontrivial (meaning not the same $u_n$) sequence equivalent to ...
4
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1answer
30 views

On finding special kinds of power series

Let $\sum a_n x^n$ be a real power series with finite positive radius of convergence $R$, then is it true that for every real number $s>0$ , we can find a real sequence $\{b_n\}$ (depending on $s$, ...
2
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3answers
32 views

If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0$ always? [duplicate]

If $E$ and $F$ are disjoint closed subsets of a metric space $(X,d)$, then is $dist~(E,F) >0?$ My attempt: Suppose $dist~(E,F)=0.$ Then $\exists~e \in E,f \in F$ such that $~\forall ...
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1answer
18 views

Logarithmically bounded function fulfills $f(n) \le \lceil m \cdot \log_b r \rceil$ for certain numbers $n,m,r$

Let $f : \mathbb N \to \mathbb N$ be a function such that $f(n) \le 1 + \log_b n$ for some base $b$ and all $n$. Now let $n \in \mathbb N$ have the property that $$ \frac{r^m - 1}{r-1} \le n < ...
2
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1answer
28 views

An inequality involving supremum and integral

Let $g$ be a positive function defined on $(0,\infty)$. Is the following inequality always true ? $$ \sup_{r<t<\infty}g(t)\leq C\int_{r}^{\infty}g(t)\frac{dt}{t}, $$ where positive constant $C$ ...
2
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2answers
62 views

Is $\int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} dx dy dz$ finite?

My question is in the title : How could I prove that $$ \int_{x=0}^1\int_{y=0}^1\int_{z=0}^1 \frac{1}{(x-y)^2 (y-z)} \ \text{d}z \ \text{d}y \ \text{d}x $$ is finite (if it is) ? Thank you by ...
2
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1answer
29 views

Symmetric function of two normal distribution implies bilinear

This question is related to my previous question which was partially answered my @MichaelHardy. Let $X$ and $Y$ be two independent standard normal random variables. Now, suppose that ...
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1answer
24 views

Does almost everywhere differentiablty imply existence of weak derivitive?

Does almost everywhere differentiablty imply existence of weak derivitive? What about the converse? If not in general maybe on compacts?
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1answer
28 views

Approximation, $C^{1}$ function

I have a question about approximation by $C^{1}$ fuctions. Let $f:[0,1] \to \mathbb{R}$ be a Lipschitz continuous function. Question Let $\epsilon>0$. Can we find a $C^{1}$ fucntion $G$ ...
4
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2answers
51 views

Existence of a solution for a nonlinear ODE on $[0,\infty)$

I'd like to prove that the solution to the following IVP exists on $[0,\infty)$. The IVP is given by $$ \begin{cases} y'(t) = y^2 \cos(t)-ye^t \\ y(0)= y_0 \end{cases} $$ where $y_0 ...
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1answer
85 views

quantum mechanics violate Bell's inequality

I have this function $$ \begin{aligned} F\big(\theta_a,\theta_b,\phi_a,\phi_b\big) = \ & – \big[\cos \theta_a \cos \theta_b \big] – \big[\sin\theta_a \sin\theta_b \sin\phi_a \sin\phi_b\big] \\ ...
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2answers
63 views

Compute the integral over the volume of a torus,

In $\mathbb R^3$, let $C$ be the circle in the $xy$-plane with radius $2$ and the origin as the center, i.e., $$C= \Big\{ \big(x,y,z\big) \in \mathbb R^3 \mid x^2+y^2=4, \ z=0\Big\}.$$ Let $\Omega$ ...
2
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2answers
80 views

Convergence of Sequence

If I know that the sequence $\{a_n\}$ converges to $a$, then to prove that the sequence $\{ca_n\}$ (for a constant $c$) converges to $ca$, I would basically want $|ca_n - ca| < \epsilon$... So, to ...
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2answers
63 views

Historical Approach to $\lim_{x \to 0} \frac{e^{\alpha x} - e^{\beta x}}{x}$, without L'Hospital's Rule

I encountered this problem, amongst others, in the slightly older Calculus textbook Piskunov's Differential and Integral Calculus when I was working with a student: Calculate the limit $$ \lim_{x \to ...