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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What can the Weierstrass therem say about an arbitrary continuous function being analytic?

Please describe the conditions and why they do not match up, as clearly not every continuous function is analytic. Thank you!
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2answers
25 views

product of uniformly continuous functions not necessarily uniformly continuous

I was asked to show that given two functions $f:\mathbb{R}\rightarrow \mathbb{R}$ and $g:\mathbb{R}\rightarrow \mathbb{R}$ which are both uniformly continuous, to show that the product ...
0
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1answer
19 views

Continuum Hypothesis?

In Kolmogorov and Fomin's Introduction to Real Analysis, there are a pair of problems which seem to be asking the reader to prove the Continuum Hypothesis. These are in Section 3, problems 12 and 13, ...
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2answers
28 views

Theorem 3.29 in Baby Rudin

Theorem 3.29 in Walter Rudin's Principles of Mathematical Analysis, 3rd ed., states that If $p>1$, then the series $$\sum_{n=2}^\infty \frac{1}{n (\log n)^p} $$ converges; if $p \leq 1$, the ...
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1answer
19 views

Fejer's theorem with Riemann integrable function

If $f$ is integrable and $f(x+), f(x-)$ exists for some $x$, then $$ \lim_{N \rightarrow \infty} {\frac{1}{{2\pi }}\int_{ - \pi }^\pi {f\left( {x - t} \right){K_N}\left( t \right)dt} } = ...
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2answers
39 views

Prove: $f(x)=e^{ax}$ is continuous on $\mathbb{R}$

Am I being fooled by how simple this statement looks? My book is currently telling me to take both $\lim_{x\rightarrow 0} f(x) =1$ and $f(x_1+x_2)=f(x_1)f(x_2)$ where $-\infty<x_1,x_2<\infty$, ...
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1answer
20 views

Differentiating both sides of an equality with respect to first variables? (Not answered)

I am proving a statement and the truth of the following proposition would help me with it. If anyone could say whether the proposition is true and give a hint how to prove it I would be very much ...
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23 views

A Riemann Integrability Question

Define $f:\mathbb{R} \rightarrow \mathbb{R}$. For any fixed closed interval $[a,b] $,$f(x) $ is $Riemann$ integrable on $[a,b].$ Proof:$\forall a,b;c,d\in\mathbb{R},a<b,c<d.$ $f (x+y) $ is ...
1
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1answer
13 views

Showing $\left \lvert \sum_{k=1}^n x_k y_k \right \rvert \le \frac{1}{\alpha} \sum_{k=1}^n x_k^2 + \frac{\alpha}{4} \sum_{k=1}^n y_k^2 $

Let $\vec x, \vec y \in \mathbb{R}^n$ and $\alpha > 0$. Show that $\left \lvert \sum_{k=1}^n x_k y_k \right \rvert \le \frac{1}{\alpha} \sum_{k=1}^n x_k^2 + \frac{\alpha}{4} \sum_{k=1}^n y_k^2 ...
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3answers
29 views

Show that if $A,B$ are measurable, $A\subset E\subset B$, and $m(A)=m(B)$, then $E$ is measurable.

Here's the full problem: Suppose $A\subset E\subset B$, where $A,B$ are measurable with finite measure. Show that if $m(A)=m(B)$, then $E$ is measurable. Here, we are dealing with measure space ...
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0answers
16 views

Constraining mathematics to a subset of $\mathbb{R}$

Let's imagine we're only using rational numbers for everything in mathematics. Problems arise quite soon when you try to calculate diagonals of squares or perhaps roots of something like $f(x)=x^2-2$. ...
2
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1answer
18 views

Real Analysis alpha Holder condition

Holder condition A function $f:(a,b)\rightarrow R$ satisfies a Holder condition of $\alpha$ order if $\alpha > 0$, and for some constant $H$ and for all $u,x \in (a,b)$, $$|f(u)-f(x)| \leq ...
2
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1answer
19 views

Cauchy construction of the real numbers.

I know that similar questions have been asked before, but I can't seem to find something that really justifies the Cauchy construction of the reals. One question that seemed to have been asked is how ...
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0answers
27 views

How to prove uniform convergence of series of functions $f_n(x) = x^n$

How do I prove that the series of functions $f_n(x) = x^n$ converges to zero on x = $[0,1)$ (without using limits)? I know I must solve for some $N \in \mathbb{N}$, but when I try to do this I cannot ...
2
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1answer
24 views

Series and comparison test

If $a_n>0$ and $\sum a_n$ diverges, what can be said about $\displaystyle \sum \frac{a_n}{1+na_n}$? I cannot prove that it is convergent or divergent. I think it is convergent for some examples ...
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2answers
101 views

Why can't consecutive irrational numbers be treated mathematically as limits?

I'm a relative newcomer to these stackexchange websites, and this post will serve as my introduction to the Mathematics stackexchange site. After perusing some of the related questions, I found these ...
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0answers
43 views

How to show that addition is continuous?

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
0
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1answer
38 views

Finding a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$.

Suppose $f,g:[a,b]\to \mathbb R$. Provide a counterexample: If $f=g^2$ and $f$ is differentiable on $[a,b]$, then $g$ is differentiable on $(a,b)$. I've been attempting to find a counterexample by ...
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1answer
31 views

Test whether $\sum_{n=1}^{\infty}\frac{\ln{n}}{n}$ converges or diverges

I am trying to solve this using an integral test, but I am unsure whether or not this is correct. Let $f:[2,\infty)\to\mathbb{R}$ be defined by $f(t)=\frac{\ln{(t)}}{t} >0\ \forall t\geq2$. Now ...
3
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2answers
39 views

Tedious undefined limit without L'Hospital $\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$

When I try to calculate this limit: $$\mathop {\lim }\limits_{x \to \frac{\pi }{2}} \,\,\frac{{\tan \,(x)}}{{\ln \,(2x - \pi )}}$$ I find this: $$\begin{array}{l} L = \mathop {\lim }\limits_{x \to ...
2
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1answer
14 views

How to identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? (measurability of function)

How can we identify $L^2(Q_T)$ and $L^2(0, T; L^2(\Omega))$? For my understanding: $Q_T:=(0, T)\times \Omega$; DEF1: $L^2(Q_T)=\{u: (0, T)\times \Omega \to \mathbb{R}, \mbox{measurable and} ...
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2answers
49 views

Limit of a headache-giving series [on hold]

I found this problem somewhere which says to find out the limit of this series, in order to prove that the limit is somewhere outside $\mathbb{Q}$. $$ x_n = \sum_{k=0}^n 2^{-k^2-k}\;,\quad \forall n ...
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2answers
37 views

How to prove this sequence is Cauchy

Show directly (from the definition) that if $$x_n=1+\frac{1}{2!} + \frac{1}{3!} + \ldots + \frac{1}{n!}\;,$$ then $(x_n)$ is a Cauchy sequence. (Hint: first show that $n! < 2^{n-1}$ for all $n$.) ...
0
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1answer
18 views

Which of the following options are correct?

$f(x),g(x)$ are defined on $[-1,1]$, $f'(0),g'(0)$ exist, $f(0)=g(0)$, and $f(x)\ge g(x)$ holds for an open interval containing $0$. Then which of the following is correct: I, $f(x)$ and $g(x)$ have ...
1
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1answer
38 views

Calculate $\int_0^1f(x)dx$

Calculate $\int_0^1f(x)dx$,where $$\ f(x) = \left\{ \begin{array}{l l} 0 & \quad \text{if $x=0$ }\\ n & \quad \text{if $x\in(\frac{1}{n+1},\frac{1}{n}]$} \end{array} \right.$$ ...
2
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1answer
34 views

Construct a Compact set of Real numbers whose limit points form a countable set

What do you think about this set? K = 0 $ \cup$ {1/n : n $\epsilon$ Natural numbers} It has one limit point which is zero, so it is countable and it is compact. Is this correct? Also, do you ...
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0answers
18 views

Applying the Implicit Function Theorem from R3 to R [duplicate]

Suppose $f:\mathbb{R}^3 \rightarrow \mathbb{R}$ is such that the Implicit Function Theorem applies to $F(x,y,z) = 0$ to determine $z = f(x,y)$, $x=g(y,z)$ and $y=h(x,z)$ in a neighborhood of a point ...
0
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1answer
17 views

A function on set involved in product of measurable sets

Let $\mathfrak{S}_1$ and $\mathfrak{S}_2$ be two families of measurable sets, and let $C\in\mathfrak{S}_1\times\mathfrak{S}_2$ be the countable union of disjoint sets, i.e. $C=\bigcup_{n=1}^\infty ...
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0answers
9 views

Multiplicative constant in inclusions of Hölder spaces

Let $\Sigma(n + \beta, L)$ for $n \in \mathbb{N}_0$, $0 < \beta \le 1$, $L > 0$ be the set of functions $f : \Omega \to \mathbb{R}$ (or $\mathbb{C}$, whatever) whose derivatives up to order $n$ ...
3
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2answers
34 views

Proof verification of compactness

Let $K$ be the set $\{0\} \cup \{1/n : n \text{ is an element of the positive integers}\} $ Prove that $K$ is compact. In my head, it seems that what they are asking in this question to prove is ...
0
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1answer
36 views

Which of the following option is true?

$\forall \varepsilon>0,\exists \delta>0$ such that $|f(x)-f(x_0)|>\varepsilon$ whenever $|x-x_0|>\delta$ is equivalent to which of the following? $1.f$ is unbounded. ...
0
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2answers
14 views

Uniform convergence - definition / notation clarification

My professor gave the following definition in class for uniform convergence: $(f_{n}: A \subset \mathbb{R}^{k} \rightarrow \mathbb{R}^{l})_{n=1}^{\infty}$ converges uniformly to $f$ on $A$ if and ...
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0answers
19 views

I need a finite lower bound for this functional, or to prove that one does not exist.

Let $0\leq g < \kappa$, $\gamma>0$ and let $f_1,f_2, S$ be arbitrary functions of r, with $f_1,f_2\geq 0$ I'm looking for a lower bound on the functional $\mathcal{E} = \frac{1}{2} ...
2
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0answers
21 views

Does $\mu_k(U \cap \mathbb{R}^k)=0$ for all affine embeddings of $\mathbb{R}^k$ in $\mathbb{R}^n$ imply $\mu_n(U)=0$?

Is the following true: We write $\mu_n$ for the Lebesgue measure on $\mathbb{R}^n$. Let $U \subset \mathbb{R}^n$, $U$ measurable and $k \leq n$. Say for every affine embedding $i \colon \mathbb{R}^k ...
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0answers
13 views

Uniform convergence on subintervals

(a) Fix a positive integer $M$ and let $\{f_{n} : [0, M]\rightarrow \mathbb{R}\}$ be a sequence of functions. Suppose that $f_{n}\rightarrow f$ pointwise on $[0, M]$ and that $f_{n}\rightarrow f$ ...
1
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1answer
18 views

Claim about differentiability from rn to r1

I'm reading over my notes and there is a claim that states: if $f:\mathbb{R}^n \rightarrow \mathbb{R}$ satisfies $|f(x)| \le |x|^2$ on $|x| < \delta$, for some $\delta > 0$, x must be ...
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Properties of a continuous function

Let $f:\mathbb R\rightarrow \mathbb R$ be a continuous and bounded function.Then which of the following are true: a.$f $ has to be uniformly continuous. b.$\exists $ an $x \in \mathbb R$ such that ...
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27 views

My attempt to prove an inequality get stuck——————where do I go wrong?

Hi, there. Bellow is my attempt. I don't know if I have gone in the wrong way and I am stuck. My attempt: Using Green's representation formula, $u(y)=\int_{\partial \Omega}u \frac{\partial ...
2
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1answer
21 views

Convergence of $\sum(-1)^k\frac{(\ln k)^p}{k^q}$ where $p,q$ in positive $\mathbb{R}$

For any $p, q$ in positive $\mathbb{R}$ $$\sum_{k=2}^{\infty}(-1)^k\frac{(\ln k)^p}{k^q}$$ I want to Use alternative series test for convergence but I'm struggling to verify that $\frac{(\ln ...
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0answers
19 views

Problem about Riemann integrable function's uniform convergence

I have no idea of how to answer the following question. It seems that the function is recurrence.
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1answer
21 views

Proof of a limit formula

If $h(x) = f(x)/g(x)$ $lim(x->b) f(x) = L$ $lim(x->b) g(x) = M$ Prove that $lim(x->b) h(x) = L/M$ Sorry for the terrible latex. ONLY FORMAL PROOFS! For every $\epsilon > 0$ Since ...
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1answer
9 views

Hardy-Littlewood maximal operator

Consider the centered Hardy_littlewood maximal operator $$ \mathcal{M}f(x):= \sup_{r>0} \frac{1}{|B(x,r)|} \int_{B(x,r)} |f(y)| \text{d}y $$ and the uncentered $$ Mf(x):= \sup_{r>0, ...
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1answer
10 views

Deduce the Bolzano-Weierstrass Theorem from the Heine-Borel Theorem

I'm working through a proof of the above that goes something like If $I$ is a compact interval, suppose toward a contradiction that there exists an infinite real sequence $\{x_n\}$ in $I$ without ...
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0answers
15 views

Finding a formula in terms of matrix elements of linear transformation

I have been given this problem: Let $T:\mathbb{R}^1 \rightarrow \mathbb{R}^n$. Find a formula for $\|T\|$ in terms of the matrix elements of $T$. Prove your formula, and illustrate it with an ...
2
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0answers
20 views

If $f$ is increasing, then for all $n\in\mathbb{N}$ there exists $P_n$ : $U(f,P)-L(f,P) \leq (b-a)/n$

I've already proven that, if $f:[a,b] \to \mathbb{R}$ is continuous and increasing, with $a,b\in \mathbb{R}$, then $$U(f,P) - L(f,P) = \sum_{i=1}^{n}\left[ f(x_i) - f(x_{i-1})\right](x_i - x_{i-1})$$ ...
1
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2answers
43 views

Continuity of a mapping $C\to C^2$, $C$ being the Cantor set

I will denote the Cantor set as $C$. We have proved earlier that every $x\in C$ can be uniquely written in a ternary representation $x=0.a_1a_2a_3...$ where all the $a_i \in \{0,2\}$. Now we consider ...
0
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0answers
13 views

I want to compute the Fourier Transform of $g(x) = (1+e^{a x})^{1/a} \mathbf{1}_{x<0} e^{-x}$

I would like to compute the fourier transform of $g(x) = (1+e^{a x})^{1/a} \mathbf{1}_{x<0}\ e^{-x}$ Question 1: Does the transform exist in any sense?. The sufficient condition ...
0
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0answers
15 views

If $\limsup_{t\to \infty} \int_{0}^{t}Tr(A(s))ds = \infty$ then $\limsup_{t\to \infty} |x(t)|=\infty$

For a homogeneous linear system of differential equations: $x'=Ax$ : Suppose that $\limsup_{t\to \infty} \int_{0}^{t}tr(A(s))ds = \infty$ ($tr(A):=$ trace of the matrix A). Then there exists solution ...
0
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1answer
49 views

Limit Delta-Epsilon proof

Prove $\lim_{x \to a} 2x = 2a$ Using the formal proof, not informal. So we know $2|x - a| < \epsilon$ We need to find some $\delta$ We only need to prove there IS SOME $\delta$ right? Only ...
0
votes
2answers
49 views