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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Associated Legendre polynomials

The associated Legendre ODE is given by $$ \left( (1-x^2) f'(x) \right)' - \frac{m^2}{1-x^2} f(x) = \lambda f(x)$$ The eigenfunctions have certain properties that I would like to understand by ...
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3answers
39 views

The sequence of improper integrals of the form $\int\frac{dx}{1+x^{2n}}$

Let $n\in\mathbb N$ ($n>0$), and define the $n$th integral in the sequence $I$ to be $$I_n = \int_{-\infty}^{\infty}\frac{1}{1+x^{2n}}dx.$$ Evaluating such integrals, especially for small $n$, is ...
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15 views

A basic non-autonomous O.D.E question

Consider the following non-autonomous O.D.E $$ \dot{x}(t) = h(x(t),g(t))$$ such that $h(.,.)$ is continuous but $g(.)$ is discontinuous(step function). Does the solution exist here ? I don't think ...
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1answer
21 views

Finite coloring of an interval

Two real functions, $f$ and $g$, are defined on the interval $[-1,1]$. Each point $x$ in the interval is colored in one of 3 colors: Red - if $f(x)>g(x)$ Blue - if $f(x)=g(x)$ Green - if ...
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55 views

Convergence of summable sequences

If $(a_n)$ is a sequence such that $$\lim_{n\to\infty}\frac{a_1^4+a_2^4+\dots+a_n^4}{n}=0.$$ How do I show that $\lim_{n\to\infty}\dfrac{a_1+a_2+\dots+a_n}{n}=0$?
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5answers
60 views

Every sequence in $\mathbb{R}$ has a monotonic subsequence

I have trouble with this kind of infinite construction in topology. Can someone check my proof is sound? Let $s$ be a sequence in $\mathbb{R}$. Then $s$ has a monotonic subsequence. There are two ...
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28 views

Properties of the Fourier transform

Suppose $g,h \in L^1(\mathbb{R} / 2 \pi)$ with $g(x)=h(nx)$, $n \in \mathbb{Z}$. I want to show that $$ \widehat{g}(kn)= \widehat{h}(k), \\ \widehat{g}(l)=0, l \not\equiv 0 \ \text{mod} \ n.$$ I ...
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1answer
13 views

Young's inequality implies $L^p$ convergence of convolution

I am reading a material which states: If $f_n \to f$ in $L^1(\mathbb{R})$, $g \in L^P(\mathbb{R})$. Then $f_n*g \to f*g$ in $L^p(\mathbb{R})$ by Young's inequality. But I cannot see why Young's ...
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2answers
18 views

Continuous piecewise smooth curve

I cannot understand the definition of $\tilde d(p_1,p_2)$ here? Can anyone please explain it clearly?
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1answer
49 views

Finding points that satisfy $f(a) = \sup f(x)$

Choose positive real numbers $\alpha_1,\ldots,\alpha_n$, $n$ such that $\sum_{i=1}^n \alpha_i = 1$ and let $$f: [0,\infty)^n \to \mathbb R$$ $$x=(x_1, \ldots, x_n) \mapsto x_1^{\alpha_1} \cdots ...
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1answer
45 views

Application of Rolle's Theorem and differentiation

Suppose $f: \mathbb{R}\rightarrow \mathbb{R}$ is differentiable with $f(0)=f(1)=0$ and $\{x:f'(x)=0\}\subset \{x:f(x)=0\}$. Show that $f(x)=0$ for all $x\in [0,1]$. My Work: By Rolle's Theorem ...
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0answers
17 views

Surjective bilinear map

Let $Q$ be a CONVEX quadrilateral in $R^2$ with vertices $a_1,a_2,a_3,a_4 \in R^2$. Consider the bilinear map $f: [0,1]^2 \to Q$ $$f(x,y)=a_1+(a_2−a_1)x+(a_4−a_1)y+(a_1+a_3−a_2−a_4)xy$$ Note that $f$ ...
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23 views

Show that there is $f\in L^1(X,\mu)$ with $P(f)<\infty$ and $P(f_n-f)\to 0$ as $n\to\infty$

Could you please help me solving this old prelim problem. Any hints are appreciated
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1answer
37 views

Proving existence of a linear functional

Let $(X, \| \cdot \|)$ be a normed space, and let $A, B ⊂ X$ be disjoint convex sets such that $B$ is closed and $A$ is compact. Prove that there exists $\varphi ∈ X^*$ such that $$\sup_{a\in A} ...
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1answer
23 views

Mean value inequality geometrical interpretaion

The mean value inequality theorem Let U be an open interval in $\mathbb{R}$. Suppose that $K \ge 0$ and that, $a,b \in U$ with $b>a$. If $f : U \rightarrow \mathbb{R}$ is differentiable with ...
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5answers
80 views

Analysis: Prove divergence of sequence $(n!)^{\frac2n}$

I am trying to prove that the sequence $$a_n = (n!)^{\frac2n}$$ tends to infinity as $ n \to \infty $. I've tried different methods but I haven't really got anywhere. Any solutions/hints?
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2answers
32 views

Taking derivatives of the implied function - from the implicit function theorem,

I showed that the relation $$f(x,y)=e^x - e^y + xy = 0$$ defines near (0,0) an implicit function y=$\phi (x)$, since the $1x1$ block, $\frac{df}{dy}$, evaluated at (0,0) gives -1, which is non-zero - ...
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25 views

About fractional iterations and improper integrals

Let $g(x,0) = x$ and $g(x,t+1) = g(x,t) - \dfrac{1}{g(x,t)}$ for every real $t$. From the fact \begin{align} ...
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29 views

Two disjoint connected and bounded open sets in the plane that shares the same boundary

In $\mathbb{R}^2$ with std. topology I want to exhibit two open sets that are connected, bounded and disjoint but that have a common boundary. My attempt: Since both my sets need to be bounded, my ...
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0answers
16 views

Uniqueness of an embedding theorem for Real differential fields

I will follow a preliminary exposition for the problem in question, which will essentially follow the format on http://www4.ncsu.edu/~singer/papers/model_diff_fields.pdf [pg. 87]: Let $K$ be a real ...
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0answers
28 views

Anyone knows a Good Textbook in Numerical PDES

I am planning on taking a course on numerical PDEs next semester. The course covers the following topics listed below. I am looking for a good book that covers these topics (or at least most of them). ...
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59 views

Using $\epsilon$ and $\delta$ to prove that $\lim_{x \to 0}\frac{\tan x}{x}=1$

I tried $$ \left | \frac{\tan x}{x}-1 \right | <1+\left | \frac{\sin x}{x} \right |\left | \frac{1}{\cos x} \right |<1+\frac{\delta^2}{2}\left | \frac{1}{\cos x} \right | $$ and I failed to ...
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1answer
27 views

Integral with a compact supported function $0$ indicates the $L^2$ function $0$ almost everywhere

Suppose we have $f\in L^2([0,1])$,and for every $\varphi\in C_{0}^{\infty}((0,1))$, we have $$\int_0^1 f(x)\varphi(x)dx=0$$ Then how can I show $f=0$ a.e? I know when $f\in C^0([0,1])$ the results ...
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1answer
34 views

Finishing a proof: $f$ is injective if and only if it has a left inverse

I've already done a lot of searching (in particular: https://www.proofwiki.org/wiki/Injection_iff_Left_Inverse) to try to prove this statement: $f: A \to B$ is injective if and only if it has a ...
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1answer
28 views

Ordered Field: $|x|\le y$ iff $-y\le x\le y$

I had a question regarding this part of a theorem that describes the inequalities of the absolute value function for order field $\mathbb{F}.$ Here is the theorem: Theorem: Let $\mathbb{F}$ be an ...
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3answers
29 views

proof of chain rule

Is my proof correct? show: $(g\circ f)'(x_0)=g'(y_0)f'(x_0)$ Since $f'(x_0) = \lim_{x\to x_0} \frac{f(x)-f(x_0)}{x-x_0}$ and Since $g'(y_0) = \lim_{y\to y_0} \frac{g(y)-g(y_0)}{y-y_0}$ Multiply ...
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21 views

Fourier series - Understanding an equality

Why is this equality true: $$\left\langle {f,g} \right\rangle = \sum\limits_{n = - N}^N {\hat{f}(n)\hat{g}(n)}$$ where $$f = \sum_{n=-N}^N c_n e^{int}, g=\sum_{n=-N}^N d_n e^{int} $$ and ...
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1answer
33 views

How do I prove this derivation?

I hope you can help me with this one because I seem to not quiet get a start here :/ Lets say we got a $b\in\mathbb{R}_{\gt 0}$ and a $y\in\mathbb{R}$ and we define $b^y:=\exp\left(\ln b \cdot ...
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2answers
20 views

Show that $\lim_{n\to\infty}\sum_{k=1}^n\bigl|k\bigl(f\bigl(\frac{1}{k}\bigr)-f\bigl(-\frac{1}{k}\bigr)\bigr)-2f'(0)\bigr|$ exists

Suppose $f\in C^3[-1,1]$, show that $$\lim_{n\to\infty}\sum_{k=1}^n\left|k\left(f\left(\frac{1}{k}\right)-f\left(-\frac{1}{k}\right)\right)-2f'(0)\right|$$ exists. I realized that ...
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1answer
74 views

Does $\int_{0}^{1}x^nf(x)\, dx=0$ imply that $f=0$ a.e. without assuming $f \in C[0,1]$?

Suppose that $f \in L^{1}[0,1]$ and $\int_{0}^{1}x^nf(x)\, dx=0$ for $n=0,1,2,\dots$ Does that imply that $f=0$ a.e.? I think that there will be a counterexample but it is hard to find out.
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1answer
25 views

differentiabilty implies continuity (analysis)

Is my proof correct? We need to show that if $f$ is differentiable at $x_o$, then it is continuous at $x_o$ i. e. $$\forall \epsilon >0, \exists \delta >0 \text{ s.t. } ...
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2answers
84 views

Probabilistic techniques, methods, and ideas in (“undergraduate”) real analysis

As the book Probabilistic Techniques in Analysis by Richard Bass shows, nowadays techniques drawn from probability are used to tackle problems in analysis. The mentioned book presents a survey of ...
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93 views

Find the values of the positive integers $n$ such that: $\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$ is positive integer

My question is as follow: Find the values of the positive integers $n$ such that: $$\frac{(-\sqrt{3}+2)^{n+1}+(\sqrt{3}+2)^{n+1}}{4n+3}$$ is positive integer. I can see that for $n=1$ (among some ...
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2answers
37 views

Solving the integral equation $y(x) = 3 + 2\int_1^x t y(t) dt $ by reducing it to a differential equation

Solve the integral equation $$y(x) = 3 + 2\int_1^x t \ y(t) \ dt $$ First I solved for the integral equation. Then I'm told to differentiate and I get $${dy \over dx} = 2 x y(x) $$ Then I ...
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0answers
38 views

Is the inverse function continuous at a fixed point?

Show that $f:I=(-1,1) \rightarrow \mathbb{R},$ it follows that $$ f(x)=\begin{cases} \quad1-x & \text{ as } -1<x\leq 0, \\ \frac{{x}^{-1}+ \lfloor {x}^{-1}\rfloor}{1+{x}^{-1}+\lfloor ...
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1answer
36 views

Why can't the definition of convergence be alterted to this one?

I am trying to find out of a seqence with the following property is convergent: Let $(r_n)$ be a sequence of real numbers. Suppose there is a number $r\in\mathbb{R}$ such that for any ...
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1answer
36 views

sequence defined by norm

Let $(u_n)_n$ be defined by: $\quad \begin{cases}u_1=1 & \\ \\ u_n=\left( \sum\limits_{k=1}^{n}u_{k}\right)^{\frac{1}{2}} & \end{cases} $ Show that $u_{n}\to +\infty$ and $u_n \simeq ...
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2answers
35 views

Subscript underneath formula in Latex [on hold]

How to type subscript underneath formula $\bigcap$ in Latex? For example, I want to type u $\in$ $\mu$ under it.
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137 views

Proof of expression with integrals

I have had trouble proving the following expression. Do you have any hints to help me? Let $f:[a,b]$ be an integrable function for which $$\int_a^bf(x)dx=6$$ Prove that there exist $t_1,t_2\in(a,b)$ ...
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1answer
27 views

Limit sup and inf hint

I have problem in finding the Limsup and liminf for the following sequences. Any hint pls? $(s_n) = [1-r^n]\sin \frac{n\pi}{2}$ and $(s_n) = [(-1)^n + 1]n^2$.
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1answer
49 views

If a sequence $(f_n)$ converges in $L^2$, then $g'(x)\int_0^x f_n(t)\,dt$ converges in $L^1$

The first: Suppose $g$ is increasing and differentiable on $[0,1]$. For every $f\in L^2(0,1)$ define $f^*(x)$, for $x\in [0,1]$, by: $$f^*(x)=g'(x)\int_0^x f(t)\,dt .$$ If $f_n\to f$ in $L^2(0,1)$, ...
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32 views

Looking for a special kind of injective function

Does there exist an injective function $f:\mathbb R \to \mathbb R$ such that for every $c \in \mathbb R$ , there is a real sequence $(x_n)$ such that $\lim\big(f(x_n)\big)=c$ but $f$ is neither ...
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2answers
88 views

How can we write this in math?

I am given a function $f: X \times Y \times Z \times \mathbb N \to [0, \infty)$. There exists a $x \in X$ such that for any $y \in Y$ and any $\epsilon > 0$, there exists $n_\epsilon$ such that ...
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0answers
32 views

Looking for a partial converse of Rolle's theorem

Let $f: [a,b] \to \mathbb R $ be a continuous function differentiable in $(a,b)$ such that $f(b)=0$ and for some $c \in (a,b) , f'(c)=0$ ; then under what additional conditions can we conclude that ...
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1answer
47 views

Evaluate this infinite product involving $a_k$

Let $a_0 = 5/2$ and $a_k = a_{k-1}^{2} - 2$ for $k \ge 1$ Compute: $$\prod_{k=0}^{\infty} 1 - \frac{1}{a_k}$$ Off the bat, we can seperate $a_0$ $$= -3/2 \cdot \prod_{k=1}^{\infty} 1 - ...
4
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0answers
49 views

Seperating single integral into an double integral.

Please refer to : How to prove that $\int_{0}^{\infty}\sin{x}\arctan{\frac{1}{x}}\,\mathrm dx=\frac{\pi }{2} \big(\frac{e-1}e\big)$ The answer by @Venus. What is the procedure in converting that ...
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0answers
29 views

Interpretation of parametrization

Let $f(t)=(x(t),y(t))'$ for $t\in[0,1]$, represents a parametric function. Let us consider a parametric equation (straightline) joining two points $a$ and $b$ in 2-dimension: $$f(t)=a(1-t)+bt.$$ ...
2
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2answers
63 views

Computing the limit of an alternating series,

I am looking at the series $$ \sum_{n=1}^\infty\frac{(-1)^n}{n}.$$ This series converges (conditionally) by the alternating series test. How can I compute its limit, which is equal to -log(2)? a) ...
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1answer
40 views

Ordered Field $\mathbb{F}$ Corollary Proof

I wanted to check my proof for a corollary on ordered fields $\mathbb{F}$. Here is the corollary: Corollary: Let $\mathbb{F}$ be an ordered field and $a\in\mathbb{F}.$ If $a>0$, then ...
2
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1answer
43 views

On a $\epsilon$-$n$ proof of a limit of a sequence of functions.

Put $\delta_n = 2^{-n}$. To each positive integer $n$ and each real number $t$ corresponds a unique integer $k = k_n(t)$ that satisfies $k\delta_n \le t< (k+1) \delta_n$. Define $$ \psi_n(t) = ...