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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32 views

Showing that the remainder term in Taylor's Theorem Converges to Zero

On pg. 110 of Rudin's Principles of Mathematical Analysis, it is shown that if $f$ is a real function on $[a, b]$ with $f^{(n)}(t)$ existing for every $t \in (a,b)$, then there exists some $x \in (a, ...
3
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1answer
51 views

Proof of Stirling's Formula using Trapezoid rule and Wallis Product

I need a proof of stirling's formula which uses the riemann's sum and trapezoid approximation to come up with $ \frac {n!}{(n/e)^n \sqrt n}$ $ \rightarrow C$ where $C$ is derived from Wallis product. ...
1
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1answer
36 views

Algebraic number spaces

While studying about Vector spaces and subspaces I came across the following question:- $Q.$ Do $algebraic$ numbers form a subspace of the vector space $\Bbb R$? According to my knowledge of $...
-1
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1answer
34 views

help with real analysis [closed]

Let $S \subset \mathbb{R}$ be nonempty. Show that if $u= \sup S$, then for every number $n$ belong to $\mathbb{N}$ the number $u -\frac{1}{n}$ is not an upper bound of $S$, but the number $u + \frac{1}...
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0answers
45 views

About limit of a function

Suppose $\lim\limits_{(x,y)\to (0,0)}\frac {f(x,y)}{g(x,y)} =0$. But if we assume $(x,y)\to (0,0)$ along the curve $f(x,y)=g(x,y)$, then limit becomes equal to 1. How to get rid of this kind of things?...
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0answers
19 views

Derivative of quadratic form involving singularity

This might be a silly question, but i have been really curious about the following: Consider the following function seen thru a single variable, say $\alpha$: \begin{equation} f(\alpha) = \mathbf{x}^...
9
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1answer
245 views

What is the cardinal of non measureable set?

we know that $|P( \Bbb{R} )|=|L (\Bbb{R} )|$ ( $L (\Bbb{R} ) $ is the set of all Lebesgue measureable set ) note that $L (\Bbb{R} ) \subsetneq P( \Bbb{R} ) $. What is the cardinal of non ...
3
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2answers
36 views

Fundamental theorem of calculus for distributions

I want to show that if $f\in C([0,1])$ and the distributional derivative $f'$ on $(0,1)$ is in $L^1((0,1))$, then $$f(1) - f(0) = \int_0^1 f'(x)\,dx$$ I am having a lot of trouble getting started. ...
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3answers
83 views

Find $f'(x)$in terms of $f(x)=|\cos(x)|\sqrt{1-\cos(x)}$

I am trying to solve the following exercise : Let $f$ be the function defined by : $$\forall x\in]0,\pi[\;\;\;\;\; f(x)=|\cos(x)|\sqrt{1-\cos(x)}$$ calculate $f '(x)$ in terms of $f(x),$ for all $x\...
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2answers
74 views

Alternatives to Chapters 8-10 of Rudin's PMA

S.E advisers, I have been hearing that the chapters on multi-varaible analysis in Rudin's PMA are almost nothing like previous insightful chapters in the single-variable analysis, and I verified ...
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0answers
29 views

proving the axiom of addition and multiplication for real numbers $a$,$b$ and $c$

I have learnt about the axioms of multiplication and a few of addition of real numbers but I still have problems with proving the uniqueness of the equalities (i) $a+b+c=a+c+b=b+a+c=b+c+a=c+a+b=c+b+...
3
votes
1answer
37 views

Squeeze fractions with $a^n+b^n=c^n+d^n$

Let $0<x<y$ be real numbers. For which positive integers $n$ do there always exist positive integers $a,b,c,d$ such that $$x<\frac ab<\frac cd<y$$ and $a^n+b^n=c^n+d^n$? For $n=1$ this ...
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2answers
40 views

Additive functions and measure theory

Key reference is the following: Hamel basis and additive functions Let's investigate real-valued functions $f(x)$ with the following (additive) property for all $\,a,b$ : $$ f(a+b)=f(a)+f(b) $$ It ...
0
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0answers
30 views

Prove a function vanishes almost everywhere

Let $f$ be a Borel measurable function on the real line.Show that if $\int^{b}_{a} f(x) dx=0$ for all rationals a,b such that $\infty<a<b<\infty$ then $f(x)=0$ almost everywhere. I want to ...
2
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3answers
63 views

Use $\delta-\epsilon$ to show that $\lim_{n\to\infty} a^{\frac{1}{n}} = 1$?

Hope this is a meaningful question, but I'm curious if is possible to show that $$\lim_{n\to\infty} a^{\frac{1}{n}}=1, \text{where }a>0$$ using $\delta-\epsilon$ directly or other methods. One ...
1
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2answers
23 views

Sequence of integrable functions on $[0,1]$ with a certain bound on the $L_1$-norm converges to 0 a.e.

Given a sequence of integrable functions $f_n$ on $[0,1]$ that satisfy $\int_0^1 |f_n| dx \le 1/n^2$, I'm trying to show $f_n \to 0$ a.e. I was considering the set $A \subset [0,1]$ which consists of ...
0
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0answers
52 views

Self-learning measure theory for my background/need

I am not a math student, but I have taken courses in Calculus, Vector Algebra, Fourier and Laplace transforms, linear Algebra, elementary probability and stats while pursuing CompSci. I work as a ...
2
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1answer
24 views

Relation between pointwise convergence and convergence in measure.

Let $f_{n} \rightarrow f$ almost everywhere with $f$ integrable. Show that $\int |f_{n}-f|\rightarrow 0$ if and only if $\int |f_{n}|\rightarrow \int |f|$. Does this result still hold if we assume $f_{...
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2answers
43 views

Estimate $\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x$ without spherical coordinates.

Is it possible to estimate the following Lebesgue integral ($\|\cdot\|$ is the 2-norm) $$\int_{\|x\|\ge\delta}\frac1{\|x\|^{d+1}}\mathrm d x, \, x\in\Bbb R^d$$ in terms of $\delta$ when $\delta\to 0$? ...
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3answers
56 views

find $f\in L^2([0,\pi])$ such that its $L^2$ distance to $\sin(x)$ and $\cos(x)$ are both bounded by specific constants

I want to find all $f\in L^2([0,\pi])$ such that $$ \begin{align} \int_0^\pi\lvert f(x)-\sin(x)\rvert^2\,dx &\le \frac{4\pi}{9}\\ \int_0^\pi\lvert f(x)-\cos(x)\rvert^2\,dx &\le \frac{\pi}{9}\\ ...
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0answers
16 views

Does there exists a notion of rate of convergence for ODE?

Suppose we wish to solve $$\dot x = f_i(x)$$ Subjected to $x(0) = x_0 \in \mathbb{R}$ where $f_i$ is some collection of continuous ($C^1$ or Lipschitz) vector field Is there an objective way to ...
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1answer
31 views

Finding a delta for the greatest integer function given an epsilon = 1/2

I'm having trouble with the following problem. Given the standard greatest integer function $\lfloor x \rfloor = int(x)$ where $ \lfloor x \rfloor $ returns the greatest integer less than or equal to ...
0
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2answers
30 views

inf, sup, max, min for $\bigcap_{n \in \mathbb N}\left[-\frac{1}{3^n},4+\frac{1}{2n}\right)$

For the following set do I have the inf, sup, min, max correct? $\bigcap_{n \in \mathbb N}\left[-\frac{1}{3^n},4+\frac{1}{2n}\right)$ $\text{inf}S=0$ $\text{sup}S= 4$ $\text{min}S=0$ $\text{...
1
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1answer
25 views

Local normal convergence equivalent to compact normal convergence

Let $X$ be an open subset of $\mathbb{R}^m$ and let $f_n\colon X\to \mathbb{C}$ be complex-valued functions. Then one has the following two notions: $\textbf{1.}$ The series $\sum\limits_{n=0}^{\...
0
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0answers
37 views

Convergence of the polygonal approximation of a differentiable 3-space curve

The following is a question taken from do Carmo's Differential Geometry of Curves and Surfaces. I believe I have an alternative method of proof to the one he suggests in the hints, but I would just ...
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2answers
76 views

Real Analysis, Folland Theorem 3.27 Properties of functions of Bounded Variation

Background Information: Taking $a = -\infty$ and considering the total variation as a function of $b$. To with $F:\mathbb{R}\rightarrow \mathbb{C}$ and $x\in\mathbb{R}$, we define $$T_F(x) = \sup\{\...
0
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1answer
47 views

I want to prove that $C_0(X)$ is Banach

Let $X$ be locally compact Hausdorff space. I'm trying to prove that $$ C_0(X)=\{ f:X\to \mathbb{C} \; | \; f \text{ is continuous and }\forall \epsilon>0 \; \exists K(\text{compact}) \subset X \...
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0answers
24 views

Improving inequality $(\int X(x) Y(x) \,dx) \leq (\int |X(y)| \,dy) Y_{\max}$

Want to improve the following inequality: $(\int X(x) Y(x) \,dx) \leq (\int |X(y)| \,dy) Y_{\max}$ Looking to replace $Y_{\max}$ with something that will give a tighter bound. Everything else needs ...
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1answer
37 views

Determining Class of a general Borel measure

Let $(X, \mathcal{T})$ be a topological space, and $\Sigma = \Sigma(\mathcal{T})$ the $\sigma$-algebra of Borel sets (that is, the $\sigma$-algebra generated by $\mathcal{T}$). In Real Analysis and ...
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1answer
26 views

Prove that a strictly increasing function with Intermediate value property is continuous

Definition: A real function $f$ has the intermediate value property on an interval $I$ containing $[a,b]$ if $f(a) < v < f(b)$ or $f(b) < v < f(a)$; that is, if $v$ is between $f(a)$ and $...
2
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0answers
32 views

How to solve this Sturm Liouville problem?

$\dfrac{d^2\phi}{dx^2} + (\lambda - x^4)\phi = 0$ Would really appreciate a solution or a significant hint because I could find anything that's helpful in my textbook. Thanks!
4
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2answers
90 views

How to prove $ \lim_{n \rightarrow \infty } \frac{1}{n} \int_0^1 \ln \big( 1 + e^{nf(t)} \big) dt = \int_0^1 f^+(t) dt $

Let $ f \in L^1[0,1]$ be a real valued Lebesgue integrable function on $[0,1]$. Prove that $$ \lim_{n \rightarrow \infty } \frac{1}{n} \int_0^1 \ln \big( 1 + e^{nf(t)} \big) dt = \int_0^1 f^+(t) dt ...
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0answers
17 views

Compact set intersects finitely many elements of an open set covering of a bounded open set prove [closed]

I saw @MISC {1697170, TITLE = {Compact set intersects finitely many elements of an open set covering of a bounded open set}, AUTHOR = {Hua (http://math.stackexchange.com/users/269509/hua)}, ...
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3answers
52 views

Intermediate value property with no continuity

Definition: A real function f has the intermediate value property on an interval I containing [a,b] if f(a) < v < f(b) or f(b) < v < f(a); that is, if v is between f(a) and f(b), there is ...
0
votes
1answer
65 views

Why is this function smooth on the coordinate axis

Consider the function $$f(x,y):=\sqrt{x^2+xy+y^3}, \quad x,y \geq 0.$$ It is claimed that this function is smooth except at the origin. I am wondering why this function is not smooth at (0,0) in the ...
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1answer
65 views

Extreme points of the unit ball of the space $c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\}$

I want to prove that all "closed unit ball" of $$ c_0 = \{ \{x_n\}_{n=1}^\infty \in \ell^\infty : \lim_{n\to\infty} x_n = 0\} $$ do not have any extreme point. Would you please help me? (Extreme ...
0
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2answers
91 views

Evaluating $\lim_{n\to\infty}\int_0^n(1-(x/n))^ne^{x/2}dx$

$$ \mbox{How to compute}\quad \lim_{n \to \infty}\,\,\int_{0}^{n}\left(1 -{x \over n}\right)^{n} \,\mathrm{e}^{x/2}\,\,\mathrm{d}x\,\,\, ?. $$ No ideas how to start this one. I see that the limit of ...
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1answer
43 views

Show that f is Riemann integrable

Let $f:[0,1] \rightarrow R$ defined by $f(x) = 2$ if $x = \frac{1}{n}$ for some $n∈ℕ$, $0$ otherwise. Determine if $f$ is Riemann integrable. My attempt: Let $ε > 0$. Construct a partition P as ...
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1answer
31 views

Poisson's equation solution explicitly

How can I write poisson's equation $\partial_{xx} u = f$ solution in 1d explicitly? I have seen somewhere I can write $u(x) = \int^{x}_{0}\int^{y}_{0} f(z) dz dy - \int^{1}_{0}\int^{x}_{0}\int^{y}...
1
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2answers
42 views

$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin y}{ny(1+n^2y^2)}ndy$ via DCT?

I'm looking to calculate these limits/integrals: $$\lim_{n\to\infty}\int_0^{\infty}\dfrac{n\sin (x/n)}{x(1+x^2)}dx$$ 2.$$\lim_{n\to\infty}\int_0^{\infty}\dfrac{\sin(x/n)}{(1+x/n)^n}dx$$ I posted ...
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1answer
58 views

Operator theory to study a difference equation

I'm not an expert in operator theory (so I'm going to be very informal sorry), but I would like to be given some advice about a problem I have. Let $f$ be a function defined in $C^{\infty}(\mathbb{R})$...
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3answers
97 views

Show that if a function is nonnegative and continuous on [a,b] then f(x)=0 for all x in [a,b] [duplicate]

Show that if $f : [a, b] \to \mathbb{R}$ is non-negative and continuous on $[a, b]$ and $$\int_{a}^{b} f(x)dx = 0$$ then $f(x)=0$, for all $x ∈ [a, b]$. I'm having difficulties in proving things to ...
0
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1answer
18 views

Proving $\mathbb{R}^2$ is not separable for this metric?

Let $d_S$ be a metric on $\mathbb{R}^2$ defined as follows $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{...
0
votes
1answer
38 views

If f is continuous and differentiable, and if $f(a) = f(b) = 0$ then prove that for a real number $α$ there is an $x ∈ (a,b)$ s.t. $αf(x) + f′(x) = 0$

If f is continuous on $[a, b]$, differentiable on $(a,b)$, and if $$f(a) = f(b) = 0$$ then prove that for a real number α there is an $x ∈ (a,b)$ s.t. $$αf(x) + f′(x) = 0$$ I know that by ...
1
vote
1answer
27 views

Compute a lebesgue integration

Given $\alpha \in (0,\infty)$, show that the function $(x,y) \rightarrow e^{-\alpha xy} \sin x$ is lebesgue integrable on $(0,\infty)\times (1,\infty)$ and compute its two iterated integrals and $\int^...
1
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0answers
30 views

Dose the pointwise limit equal to the $L^{p}$ limit?

For $1<p<\infty$ Let $L^{p}=L^{p}[0,1]$ and f be a measurable function on [0,1].If $\left\{f_{n}\right\}\in L^{p}$ where $\left\{\|f_{n}\|_{L^{p}}\right\} $ is bounded and $\lim_{n\rightarrow\...
2
votes
1answer
34 views

Prove a set to be measurable set

Let $\left\{f_{n}\right\}$ be a sequence of measurable functions on $R$ and let $f$ be a measurable function on $R$.Prove that $\left\{x\in R| \lim_{n\rightarrow \infty}f_{n}\left(x\right)=f\left(x\...
0
votes
2answers
23 views

Is the following metric topological equivalent to Euclidean metric?

Let $d_S$ be a metric on $\mathbb{R}^p$ defined as $$ d_S(x,y) = \begin{cases} || x- y|| & \text{when} \ x \ \text{and} \ y \ \text{are linearly dependent} \\ ||x|| + || y || & \text{when}\ ...
1
vote
0answers
45 views

Limit at infinity of a function series

In my researches I got stuck on two similar calculations, and I'd like to deal with them in one fell swoop. 1. I want to say that $$ \lim_{x \to \infty} \sum_{n > 1} z_n \!\!\!\sum_{\substack{d \...
1
vote
1answer
48 views

Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$ where $f\in L^{1}[0,1] \cap L^{2} [0,1]$ [on hold]

Let $f\in L^{1}[0,1] \cap L^{2} [0,1]$. Compute $\lim_{p\to 1+}\left\|f\right\|_{p}$. I think the result would be $\left\|f\right\|_{1}$,but I don't know how to prove it.