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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Properties of a particular Riemann integrable function

I don't know what to deduce about $f$ in the following question Let $f:[0,1]\to \mathbb{C} $ be continuous such that $\int_0^1x^nf(x)\ dx=0$ $\ \ \forall\ n\ge 2$.what best can be said about $f$ ?
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19 views

Continuous limits

How can I show $f:[a,b] \rightarrow \mathbb{R}$ is increasing implies it is left continuous on $(a,b)$? My attempt: Case 1: $f(a)=f(b)$ Trivial case, it is a constant continuous function so it is ...
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0answers
11 views

Existence of Riemann integrable sequence of continuous functions

Can someone help me do this problem? Let $f$ be s Riemann integrable function on $[a,b]$ not necessarily continuous.Prove that there exists a sequence $\{g_n\}_{n\in \mathbb{N}}$of continuous ...
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0answers
12 views

Can someone provide some examples to illustrate the difference between Pointwise equicontinuity and Uniform equicontinuity?

I don't know what is with the subject of pointwise and uniform equicontinuity, pretty much all the material you can find online are either: Proofs i.e. pointwise equicontinuity is uniform ...
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1answer
30 views

Calculate the limit $\lim_{x \to + \infty}\int_{\mathbb{R}} \frac{t^2}{1+t^2}e^{-(x-t)^2}dt$

I am really confused with x approaching $+ \infty$. How can I solve this limit: $$\lim_{x \to + \infty}\int_{\mathbb{R}} \frac{t^2}{1+t^2}e^{-(x-t)^2}dt$$
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25 views

Real Analysis, Folland problem 3.5.28 Functions of Bounded Variation

Background information: If $F:\mathbb{R}\rightarrow \mathbb{C}$ and $x\in\mathbb{R}$, we define $$T_{F}(x) = \sup\{\sum_{1}^{n}|F(x_j) - F(x_{j-1})|:n\in\mathbb{N},-\infty<x_0<\ldots<x_n = ...
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4answers
161 views

(Non)Existence of limits

When we say that a limit of a function does not exist in $\mathbb{R}$ (or some metric space) does it make sense to say that it might exist somewhere else? [I am trying to think along lines of ...
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28 views

Measure of the set of convergence of a series

I need some help to solve this exercise. Let $a_n$ a sequence of real numbers such that $a_n\geq 0$ for all $n\in \mathbb{N}$. Let $A$ the set $$A:=\{x\in[0,2\pi]:\sum_{n=1}^\infty ...
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1answer
13 views

How lim sups here converted to appropriate limits?

This question is from Rudin's Principles of Mathematical Analysis : Consider the sequence $\{a_n\}$: $\{\frac 12, \frac 13, \frac 1{2^2}, \frac1{3^2}, \frac 1{2^3}, \frac1{3^3},\frac 1{2^4}, ...
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1answer
25 views

Proof Directional Derivative Exists at (0,0)

Before I post this I would just like to state that I know that there is a very similar question with a very similar function but I've gone through the answer and it doesn't really help me. Consider a ...
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1answer
10 views

Prove the set of subsequential limits of ($x_t$) is closed

A sequence ($x_t$) in metric space ($X,d$). Let $S$ be the set of subsequential limits of ($x_t$) ($S$ could be empty), prove that $S$ is closed. I need to prove the problem with one of the ...
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1answer
19 views

Finding the lub and glb of set.

the question is , find lub and glb of set $S = \{ m + \frac{1}{n} \mid m,n \in \mathbb{N}\}$ my approach : clearly glb of set is $1$ because if put $m = 1$ and when $n$ tends to infinity then $m + ...
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1answer
24 views

Study convergence of $f_n(x)=\frac{\sin(nx)}{\sqrt{n}}$

$$f_n(x)=\frac{\sin(nx)}{\sqrt{n}}$$ Pointwise convergence: $$\lim_{n \rightarrow \infty } \ f_n (x)=0$$ It converges to the function $f(x)\equiv0$ Uniform convergence $$f'_n(x)=n ...
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15 views

Unconditionally convergent series

Given an integer $k\geq 0$, let $A_k=\{i_j^{(k)}:j\in\mathbb{N}\}$ be a countable set of indeces. Let $\{a_p:p\in A_k,\,k\geq 0\}\subseteq\mathbb{R}$. Denote $A=\cup_{k=0}^{\infty}A_k$. Suppose that ...
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7 views

Convergence of product of functions

Let say we have a sequence of functions $(f_ng)$ in $L^2[a,b]$, where $g$ is in $L^2[a,b]$, that converges to some function $h\in L^2[a,b]$. i.e. $f_ng\to h$ in $L^2$ as $n\to\infty$. ($f_ng$ ...
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22 views

Prove that the limit of sequence $x(n) = (2^n/n!)$ is $0$ using definition of sequence.

Generally we solve questions of this $n!$ form by finding a relation between numerator and denominator. Example - $n^2/n!$ was easily solved by using $n^2/n! <= n^2/n(n-1)(n-2)$ Is there such a ...
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1answer
14 views

Surface $x^2+y^2-z^2=4$ - Tangent plane

Give the equation of the tangent plan to the surface $x^2+y^2-z^2=4$ where $(\hat{x},\hat{y})=(2,2)$ and $\hat{z}=-2$. $(x^2+y^2)-4=z^2 \iff z=\sqrt{(x^2+y^2)-4}$ or $z=-\sqrt{(x^2+y^2)-4}$. As ...
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> example of non supperadditve function [on hold]

I need an example: 1.l-superadditive 2.monotono 3.non superadditive. please help me!
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1answer
10 views

If $f$ is differentiable at the point $\hat{x}$, show that $f$ possesses directional dirivative in all directions $v \in \mathbb{R}^n - \{0\}$.

Let $U \subset \mathbb{R}^n$ an open set, $\hat{x} \in U$ and $f : U \to \mathbb{R}$. We define the directional derivative of $f$ in the direction of $v \in \mathbb{R}^n - \{0\}$ at the point ...
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14 views

On the inequality $\displaystyle\sum_{k=1}^{n}a_k\cos(kx) < -\frac{1}{\epsilon}\left|\sum_{k=1}^{n}a_k\sin(kx)\right|$

Prove that for every $\epsilon >0$ there is a positive integer $n$ and positive numbers $a_{1},...,a_{n}$ such that for every $\epsilon < x < 2\pi - \epsilon$ we have: ...
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1answer
24 views

Partial derivatives of $x$ and $y$ are equal at $(a, b)$ and $(b, a)$

If $f(x, y) = f(y, x)$ for all $(x, y) ∈ \mathbb R^2$. Show that $\frac{∂f}{∂x}(a, b) = \frac{∂f}{∂y}(b, a)$ For all $(a,b) ∈ \mathbb R^2$. So far I think that I need to put $f(x, y) = gf(x, y)$ ...
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1answer
23 views

$f$ is differentiable at the point $\hat{x}$: $f(x)-f(\hat{x}) = h(x) \cdot (x-\hat{x})$

Let $U \subset \mathbb{R}^n$ an open set, $f : U \to \mathbb{R}$ and $\hat{x} \in U$. Show that $f$ is differentiable at the point $\hat{x}$ if and only if there exists a continuous function $h ...
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2answers
30 views

Find $f(x,y)$ integrable such that $f_x(y)$ isn't integrable

Find $f(x,y)$ integrable such that $f_x(y)$ isn't integrable, where $f_x(y)$ is in fact $f(x,y)$ while $x$ is a parameter. I thought of using $\log$ in some variation, but I think it is problematic ...
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1answer
25 views

Perfect Sets Uncountable in Metric Space

I am trying to understand the structure of the neighborhoods constructed, in Rudin's Proof. This question has been touched upon else where but it really does not go into details of the construction ...
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30 views

Why is $A=\{(x_1,x_2,…,x_n)|\exists_{i\ne j}: x_i=x_j\}$ a null set?

Why is $A=\{(x_1,x_2,...,x_n)|\exists_{i\ne j}: x_i=x_j\}$ a null set? This claim was shown in a solution I ran into, and I don't see how it holds. I try to follow the formal definition of nullity, ...
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1answer
23 views

Proving $[a,b]$ is closed by proving the complement is open

I'm using this definition: a subset $A$ of $\mathbb{R}$ is called open if $$ \forall x \in A, \exists \delta > 0: ] x - \delta, x + \delta [ \subset A. $$ Now, I need to prove that $[a,b]$ is ...
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6 views

Arrangement of Convex Discs in the plane is independent of the choice of origin?

This is the Problem 3.1 in 'Combinatorial Geometry' by J. Pach, and P. Agarwal. Problem: Prove that if C is any arrangement of convex discs in the plane, then $\bar{d}$$(C,\mathbb{R}^2)$ and ...
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14 views

sequential characterisation of limits

There is a proposition left on a 2nd year vector calculus notes provided with no proof. I always having trouble writing these kind of proof and I hope someone could provide an answer for future ...
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14 views

Two problems from Avner Friedman's PDE book.

The problems are as follow: Prove that if $Lu=0$ for any $u\in C^m(\Omega)$, then $L\equiv 0$ - that is all the coefficients of $L$ vanish identically. Prove that the assertion of the previous ...
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1answer
46 views

Limit - Applicability of L'Hopital's Rule:$\lim\limits_{x \to 0^+} \frac{xe^x}{e^x-1}$

I am required to find $\lim\limits_{x \to 0^+} \frac{xe^x}{e^x-1}$. My attempt: $\lim\limits_{x \to 0^+} \frac{xe^x}{e^x-1}$ = $\lim\limits_{x \to 0^+} e^x$ $\cdot$ $\lim\limits_{x \to 0^+} ...
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1answer
29 views

Example of set A with measure zero such that $\bar{A} = \mathbb{R}$

Example of a set A that has measure zero such that $\bar{A} = \mathbb{R}$. I can think of sets that have measure zero such as $\mathbb{N, Q}$, or the cantor set. I also know that a subset of a set ...
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5 views

Two points in a polygonal-path-connected set can be connected with a non-intersecting polygonal path

Let $X$ be polygonal-path-connected and $x,y\in X$. So $x$ and $y$ can be connected by a polygonal path $P=\bigcup_{i=1}^n L_i$ where $L_i$ is a line segment $[x_i,x_{i+1}]$. Non-intersecting means ...
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1answer
17 views

Application of Riesz Representation Theorem on $C[a,b]$

Let $f$ be a function in $C[a,b]$ norm with $\|f\|_{max}=\max_{x\in [a,b]}|f(x)|$, how to find a function $g$ which is bounded variation on $[a,b]$ for which $$\int_a^bfdg=\|f\|_{max}\text{ and ...
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1answer
25 views

How can we show completeness of real numbers from the Bolzano-Weierstrass Theorem?

How can the completeness of $\mathbb{R}$ (every Cauchy sequence in $\mathbb{R}$ converges) be proven from the Bolzano-Weierstrass Theorem? We can also use the following theorems as the ingredients: ...
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2answers
25 views

Show that continuous functions on $\mathbb R$ are Borel-measurable

Here are my thoughts. We want to show that $f^{-1}$ maps Borel sets to Borel sets. Let $f:\mathbb R\to\mathbb R$ be a continuous function and $\mathcal B(\mathbb R)$ a Borel $\sigma$-algebra. Let ...
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2answers
31 views

Infinite intersection of open sets need not be open

The following is the property of an open set: The intersection of a finite number of open sets is open. Why is it a finite number? Why can't it be infinite?
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1answer
99 views

Set with positive Lebesgue measure but no interval

This question tortures me for a while: if a set $E$ has positive Lebesgue measure, does it necessarily contain an interval? I would be truly grateful for help.
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1answer
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Trivial norm and topology

It's a well known fact that every norm in finite dimensional vector spaces are equivalent. So these norms have to induce the same topology, right? My question is If I take, for example, $\mathbb{R}^n$ ...
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2answers
63 views

Is Lebsegue Measure Translation Invariant?

I am trying to prove that the Lebsegue measure is translation-invariant. Namely, given a set $X\subseteq\mathbb{R}$, I'd like to show $X + y$ is measurable and $\mathit{m}(X + y) = \mathit{m}(X)$. ...
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2answers
49 views

Sequence of Partial Sums is Convergent

How do I show that the series $\sum_{n=1}^{\infty} \frac1{(2n-1)^{n}} + \frac1{(2n)^{3 }} $ is convergent? I'm trying to use Comparison Test but I'm having a hard time looking for a convergent series ...
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1answer
30 views

Prove that a subset of a set of measure zero has measure zero

Thm Prove that a subset of a set of measure zero has measure zero. I attempted the proof, corrections appreciated. Pf Let $A=\{x_1,....,x_N\}$ be a finite set, and let $\epsilon > 0$ be given. ...
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46 views

Does the convergence of a series $\sum_{n=1}^{\infty} x_n$ imply the convergence of the alternating series $\sum_{n=1}^{\infty} (-1)^{n} x_n $? [on hold]

True or False: If $\sum_{n=1}^{\infty} x_n$ is convergent, then $\sum_{n=1}^{\infty} (-1)^{n} x_n $ is convergent.
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562 views

Is there a way to prove this exponential inequality?

I came across this proposition while trying to prove that a function was injective: if $a>b$ then $a^a>b^b$, where $a$ and $b$ are real numbers bigger than 1 . Intuitively it (somehow) makes ...
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1answer
83 views

How to show a function is not Riemann integrable

I am wondering how I can show the following function is not Riemann integrable. Since we are on a closed and bounded interval I couldn't use that it is unbounded, etc (think) ...
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1answer
31 views

Show that $ f(x)=\sum_{n=1}^{\infty} 2^{-n} f_n(x)$ defines a continuous function on $(0,\infty)$

Let $f_n$ be a sequence of continuous functions on $(0,\infty)$ with $|f_n(x)|\le n$ for every $ x>0$ and $n\ge1$, and such that $\lim_{x\to\infty} f_n(x) =0$ for each $n$.Show that $ ...
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1answer
44 views

Suppose that $f_n \colon \Bbb{R}\to [0,1]$ for $n\in \Bbb{N}$ and that each one of these functions is decreasing

Suppose that $f_n \colon \Bbb{R}\to [0,1]$ for $n\in \Bbb{N}$ and that each one of these functions is nondecreasing. Prove that there exists a function $g\colon \Bbb{R} \to [0,1]$, a countable set ...
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0answers
16 views

how to show the equivalence of density

How to show $E$ is dense if and only if $int(\mathbb{R}- E) = \emptyset$ suggestions please. I do not see how to a direct proff
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1answer
30 views

Show that $fg$ is differentiable at $\hat{x}$ and that $(fg)'(\hat{x})= g(\hat{x})f'(\hat{x}) + f(\hat{x})g'(\hat{x})$

Let $U$ an open set in $\mathbb{R^n}$, $\hat{x} \in U$ and let $f : U \to \mathbb{R}$ and $g : U \to \mathbb{R}$ two different differentiable functions at $\hat{x}$. Show that $fg$ is ...
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0answers
13 views

Tradeoff between different formulations of partition of unity

In Wikipedia, a partition of unity of a topological space $X$ is a set $R$ of continuous functions from $X$ to the unit interval $[0,1]$ such that for every point, $x\in X$, there is a neighbourhood ...
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22 views

The convergence rate of the derivative of a sequence of function

Let $v_\delta$ be a sequence of continuous diff'able function on $(-1,1)$ and $0\leq v_\delta\leq 1$. For each $\delta>0$, assume that $v_\delta(\delta)=v_\delta(-\delta)=1$ and $v_\delta(0)=0$. We ...