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.

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29 views

Why can't solutions to Autonomous ODE intersect?

Let $f: \mathbb{R}^n \rightarrow \mathbb{R}^n $ be continuously differentiable. Why can't solutions to $x'=f(x) $ intersect? I use a proof by contradiction: Assume solutions can, and do intersect ...
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10 views

Epsilon-N method - Proof verification

Prove (using the epsilon-N method) that the sequence of numbers $\dfrac{5n^3-2}{n^3}$ converges. Calculate the limit first. First we calculate the limit: $\lim_{n \to \infty} \dfrac{5n^3-2}{n^3} ...
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0answers
15 views

Find the pointwise limit of {gn} on [0, ∞). Please help!!

Consider sequence of functions gn(x)=x\over 1+x^n over [0,\infty) (a) Find the pointwise limit of {gn} on [0, ∞). g(x)= x 0 \le x \lt x 1/2 x=1 00 x>1 Show gn(x) ...
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1answer
16 views

Show that $\{\hat {f _n }\} $ converges in measure to $f$

I want to show that if $f_n $ converges in measure to $f $ and for each $n$ $\mu ( \{x : |\hat {f _n (x)} -f_n(x) \ge \frac {1 } {n } \} )=\frac {1 } {n } $, then $\{\hat {f _n }\} $ converges in ...
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0answers
11 views

integration concerning Fourier transform of homogeneous kernel(of degree 0)

Let $m\in C^\infty(\mathbb{R}^d-\{0\})$ be homogeneous of degree zero and has mean zero on the sphere$(S^{d-1})$. Then $m$ defines a tempered distribution and $\partial_j^dm\in S'(\mathbb{R}^d)$ is ...
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0answers
7 views

First order PDE with discontinuous coefficients

I want to consider the following equation $$u_t+\mathrm{sgn}(x)u_x=0,\,\,u(0,x)=u_0(x)$$ Now if $x>0$ or $x<0$ I can use the method of characteristics to obtain $u(t,x)=u_0(x-t)$ if $x>t$ and ...
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0answers
20 views

Strange inequality

I found the inequality $\beta e - \frac{3}{2} n \ log(e+Bn)+ \frac{5}{2} \ n \ log(n) + const \cdot n \geq \frac{\beta e}{2}+ \beta n $ in a textbook,provided that either $e$ or $n$ is large. We ...
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19 views

Non-negative Matrix [on hold]

Let $M \in \mathbb{R}^{n \times n}$ be a Metzler matrix, i.e., $M_{i,j} \geq 0$ for all $i \neq j$. Consider $c \in \mathbb{R}$ and $x \in \mathbb{R}^n$. Prove that the matrix $$ \text{diag}( x_1, ...
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2answers
17 views

Constructing a subsequence

An unbounded sequence has a monotonic unbounded subsequence. This seems obvious, but I can't think of how to prove it. My attempt: Let $(x_n)$ be an unbounded sequence. Let $S_n= \{ |x_1|, |x_2|. ...
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2answers
19 views

Show $\frac{1}{|x-y|^{p}}-\frac{1}{|x|^{p}}\in L^{2}(\mathbb{R})$ for fixed real y

This is homework so no answers please Show $\frac{1}{|x-y|^{p}}-\frac{1}{|x|^{p}}\in L^{2}(\mathbb{R})$ for fixed real y where $|p|< \frac{1}{2}$. Any strategies? Attempt: I will type as I go, ...
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0answers
15 views

Finding all functions that satisfy a specific property.

This question comes from Rudin's Real and Complex Analysis text (chapter 3). Let $m$ be Lebesgue measure on $[0,1]$ and define $||f||_p$ with respect to $m$. Find all functions $\Phi$ on ...
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13 views

How to calculate Fourier Transform of logarithmic function?

Given a random variable (RV) $S$ equal to the sum of two mutually independent (RVs) $X_1,X_2$,i.e.$S=X_1+X_2$ and piece-wise probability density functions (PDFs) of $f_{X_1},f_{X_2}$ are as follow: ...
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0answers
11 views

Are the conditions for multivariable integrability the same?

For a single variable function, the function needs to have a finite number of discontinuities and must be bounded over the interval of integration for it to be Riemann integrable over that interval. ...
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1answer
19 views

For which compact sets can the size of the finite subcover be bounded?

I've been struggling to find a solution to this problem: For which compact sets can you set an upper bound on the number of sets in a subcover of an open cover. My understanding is that I need to ...
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0answers
15 views

Inequality involving absolute value.

I am working on a larger proof, but for my chosen method, I need to know if the following is true: If $b_n>0$ for all $n\in\mathbb{N}$, and $b_n\leq b_1$ for all $n\in\mathbb{N}$, then is it true ...
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20 views

Prove that this function is Borel measurable

Prove that if $s\ge 0$, $f:\mathbb{R}^n\to\mathbb{R}^m$ is continuous and $K\subset\mathbb{R}^n$ is compact, then the function $$ F:\mathbb{R}^m\to [0,\infty]\\y\mapsto H^{s}(K\cap f^{-1}(\{y\})) $$ ...
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0answers
9 views

Show that if $f $ is integrable then $\mu(|f |^{-1 } (A _n ))< \infty $

Here it is stated that if $f $ is integrable on $X $ and $A _0 = [\frac {1 } {n } , \infty ) $, $A _n = [\frac {1 } {n+1 } ,\frac {1 } {n } )$ then $\mu(|f |^{-1 } (A _n ))< \infty $ for every ...
6
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1answer
58 views

A cute limit $\lim_{m\to\infty}\left(\left(\sum_{n=1}^{m}\frac{1}{n}\sum_{k=1}^{n-1}\frac{(-1)^k}{k}\right)+\log(2)H_m\right)$

I'm sure that for many of you this is a limit pretty easy to compute, but my concern here is a bit different, and I'd like to know if I can nicely compute it without using special functions. Do you ...
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1answer
33 views

Ode with step function in the right-hand-side

I want to solve the following ODE: $$\dot{X}(t,x)=F(X(t,x))$$ where $F(x)=1$ if $x>0$ and $-1$ if $x<0$. How to treat this discontinuous right-hand-side?
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1answer
56 views

Prove (or disprove) this $\sum_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergence? [on hold]

Let $a_{n}>0$ be a sequence, and $0<a\le 1$, such that $\sum\limits_{n=1}^{\infty}(a_{n})^a$ converges. Prove or disprove $\sum\limits_{n=1}^{\infty}\dfrac{a_{n}}{n}$ is convergent. I ...
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1answer
21 views

Time derivative of operator

I have to compute, at least formally, the following derivative $$\partial_t \exp(it\Delta)f(x-ct)$$ where $\Delta$ is the Laplacian and $c$ is a constant. I know that $e^{it\Delta}$ is the Schrodinger ...
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2answers
35 views

How do I verify the solution for this problem? (Advanced Calculus)

I'm working on a problem in Rudin (Chapter 1, Exercise 1.19)... Question: Suppose $\mathbf a\in R^k, \mathbf b\in R^k.$ Find $\mathbf c \in R^k$ and $r>0$ such that $|\mathbf{x-a}| = ...
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1answer
22 views

Give an example of a nested sequence of closed but unbounded intervals which does not have a point in its intersection.

I'm finding nested intervals hard to understand and I'm really stuck on this homework question. First to make sure, a closed but unbounded interval is something like this right? $[n,\infty)$ And if ...
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1answer
18 views

Convergence of sequence of functions on Banach space

Let $\{f_{\alpha_n}\}\subset{\cal L}_2^0(\mathbb R)$ be a sequence function converging to $g$ where ${\cal L}_2^0(\mathbb R)$ is a Banach space defined by $$ {\cal L}_2^0(\mathbb ...
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0answers
14 views

Theorem about n=1 wave equation in Evans

In Evans, PDE edition 2 on p68 we have a Theorem that tells us some properties about the solution to the wave equation for $n=1$. It reads: Assume $g \in C^2(\mathbb{R})$, $h\in C^{1}(\mathbb{R})$, ...
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1answer
35 views

Total variation of $f(x)=e^{-\frac{1}{x}}\sin(\frac{\pi}{x})$

Let $f(x)=e^{-\frac{1}{x}}\sin(\frac{\pi}{x})$, with $f(0):=0$. Calculate the total variation of $f$ on $[0,1]$. It is simple to show that $f \in BV[0,1]$. However, since $f$ has infinitely many ...
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1answer
37 views

Euclidean space? (Rudin, page 23 Q 19)

Question: Suppose $\mathbf a\in R^k, \mathbf b\in R^k.$ Find $\mathbf c \in R^k$ and $r>0$ such that $|\mathbf{x-a}| = 2|\mathbf{x-b}|$ if and only if $|\mathbf{x-c}| = r$ Solution: $3 \mathbf ...
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1answer
39 views

Prove that the function f(x) = cosh(x)+ cos(x) is strictly increasing for non-negative x

I know that using the mean value theorem I should get $f'(x) =$ sinh$(x)$ - sin$(x)$, but from that on I have no ideas on how to show that $f'(x) > 0$ in the specified interval. Basic trigonometric ...
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1answer
13 views

show that $\inf \{x_n+y_n:n>N\}\ge \inf \{x_n:n>N\}+\inf \{y_n:n>N\}$

show that $\inf \{x_n+y_n:n>N\}\ge \inf \{x_n:n>N\}+\inf \{y_n:n>N\}$. I am stuck on proving this. I understand it intuitively. My approach is to first do the right part. By defining the ...
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1answer
16 views

Prove that a non-empty subset of the real numbers union its boundary set is a closed set.

Prove that the boundary set of any non-empty subset of the real numbers is a closed set, and also that a non-empty subset of the real numbers union its boundary set is a closed set. $\textbf{Prove ...
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1answer
21 views

Geometry of Riemann Stieltjes integration

We know that $\int_{a}^b f(x)dx$ represents the area bounded by the curve $y=f(x)$& the straight lines $x=a$ & $x=b$. But when we integrate $f(x)$ with respect to another function $g(x)$ then ...
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1answer
11 views

sign of derivative at zero of a continuous and differentiable function

I think this is straightforward but I cannot prove it with my rusty calculus... Suppose $f(x)$ is a continuous and differentiable functionon on $[a,b]$, $f(a)>0$, and $f(b)<0$. I want to show ...
2
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1answer
67 views

Suppose that $\lim (x_n) = x$ and that $|x_n-y_n|=1$ for all $n$ exists in $\mathbb N$.

Suppose that $\lim (x_n) = x$ and that $|x_n-y_n|=1$ for all n exists in N. Prove that there exists a subsequence of $(y_n)$ that converges to either x+1 or x-1. I understand the logic behind this, ...
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5answers
67 views

Suppose that $(x_n)$ and $(y_n)$ are convergent sequences and let un=min{xn,yn}. Prove that (un) is a convergent sequence

Suppose that $(x_n)$ and $(y_n)$ are convergent sequences and let $u_n=min${$x_n,y_n$}. Prove that $(u_n)$ is a convergent sequence I feel like this should be handled by cases but it seems to be the ...
2
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3answers
25 views

Analysis proof with metric spaces

Part a) of Theorem 2.27 in baby Rudin reads (roughly) as follows: Theorem. Let $X$ be a metric space and $E\subset X$. Then the closure of $E$ is closed. Proof. Let $x\in\bar{E}^c$. Then ...
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4answers
56 views

Real Between Rationals

Let $x$ be a real number. Show that, for any $\varepsilon>0$, there exist two rationals $q$ and $q'$ such that $q<x<q'$ and $|q-q'|<\varepsilon$ How should I approach this prove?
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1answer
14 views

Correction of Proof that if f:[0,1] is a continuous function and f(x)>2 with x being in [0,1) it is not necessary that f(1)>2

Here's how I wrote it up: To approach this consider an example of a continuous function which fails to satisfy f(1)>2 even though it satisfies f(x)>2 for x in [0,1). My counterexample is a negative ...
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1answer
20 views

the conditions for a measurable function to be the uniform limit of simple functions

In our homework we are asked to prove that, on a measurable space $(\Omega,\mathcal{F})$, every function $f:\Omega \rightarrow R, f\geq 0$ can be written as the uniform limit of an increasing limit of ...
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0answers
15 views

Mesh and Riemann Sum Proof

$\bf Question$: Suppose that f is bounded on $[a,b]$ and that there exists two sequences of tagged partitions of $[a,b]$ such that $\|\dot P\|\to 0$ and $\|\dot Q\| \to 0$ , but such that ...
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1answer
53 views

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ prove the following:

For $f: \Bbb R\rightarrow \Bbb R$ continuous and $b>0$ such that $ f(0)\neq -1$ and $\displaystyle\int_{0}^{b} f(t) \, dt=0$ Show that the equation $\displaystyle\int_{x}^{a} f(t) ...
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0answers
28 views

Method of characteristic for second order pde

Can I use the method of characteristic to solve second order pdes? For instance I canconsider the equation $$u_t+u_x=u_{xx}$$
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1answer
28 views

Determining continuity of $f$ where $f(x) = 0$ for irrational $x$ and $f(x) = 1/q$ if $x = p/q$ where $p,q$ are relatively prime integers and $q>0$

This is from Folland's Advanced Calculus, Exercise 1.3.7. This is my attempt: == First, let $x$ be rational; then $f(x) = 1/q$ is also rational. From Exercise 1.3.6, $\mathbb{I}$ is dense in ...
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2answers
19 views

finite padic number uniqueness

suppose $\sum\limits_{i=0}^{n}a_ip^i=0$ where $a_i\in \{0,1,-1,\dots,(p-1),-(p-1)\}$ and $p\geq 2, p\in N$, how to prove that $a_0=a_1=\dots=a_n=0$?
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1answer
20 views

diffeomorphisms preserve zero measure

Suppose $\Omega\subset \mathbb R^N$ is an open set and $f:\Omega\rightarrow f(\Omega)$ is a $C^1$ diffeomorphism. Show that if $F \subset \Omega$ has zero measure then $f(F)$ has zero measure. I ...
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1answer
23 views

Sequence of Measurable Functions defined on R

I'm struggling with this problem from my most recent homework assignment in measure theory. Let {$f_n$} be a sequence of measurable functions defined on $\Bbb R.$ Show that the set $E =$ {$x \in ...
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11 views

If $\mathcal{E}$ are subsets of $X$ and $A \subset X$. Show that $\mathcal{A}_X\mathcal{(E \cap}$ A)=$\mathcal{A}_A\mathcal{(E)} \cap$ A [duplicate]

If $\mathcal{E}$ is collections of subsets of a set $X$ and let $A \subset X$ be a subset. Show that the generated $\sigma$-algebra of $\mathcal{(E \cap}$ A) in $X$ =the generated $\sigma$-algebra ...
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2answers
32 views

$f(x)=1$, for every $x \in [0,1]$ if $f:[0,1]\to\mathbb R$ is continuous and $f(p)=1$ for every $p\in [0,1]\cap\mathbb Q$.

How would you approach this if I have to use the fact that "every number is a sequence of rational numbers"? Currently, I am proving this by contradiction in the following way: Let f(p)=1 for all ...
0
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0answers
12 views

Measure of intersections [duplicate]

Let $(X,\mathcal{A},μ)$ a measurable space and let $A_1,A_2,...\in \mathcal{A}$, assume that $\sum^{\infty}_{j=1}\mu(A_j)<\infty$ I want to show that ...
2
votes
1answer
17 views

Examples of measures that induce certain inclusions in the Lp spaces.

I apologize for the terribly worded title, but I didn't know how else to title this questions (which comes from Rudin's Real & Complex Analysis chapter 3 questions). The question says: For ...
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2answers
31 views

Show that if $f'_n$ converges uniformly on $[a,b]$ then $f_n$ converges uniformly on $[a,b]$.

If $f_n(x)$ is a sequence of functions differentiable on $[a,b]$ with continuous derivatives and such that $f_n(x_1)$ converges for some point $x_1$ in $[a,b]$. Without using Rudin theorem 7.17 ...