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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6
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0answers
66 views

Real analytic methods for the following integral [duplicate]

A few days back, the following integral was posted $$\int_0^1 x^x(1-x)^{1-x}\sin(\pi x)\,dx=\frac{\pi e}{24}$$ The integral was answered using complex analysis tools but I am interested in other ...
0
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0answers
20 views

Prove the combination rule for non-negative series by using comparison test

The combination rule for non-negative series is as follows: The question is, how can I prove combination rule for non-negative series by using the comparison test, namely:
1
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2answers
31 views

Real numbers intervals

Is it true that there always exists a irrational number in any open interval of real numbers? I guess yes, but I could not prove it. Could any help me with this? Thanks a lot!
1
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1answer
22 views

Constant raised to the power of an even or odd function

Suppose that $a$ is a positive real number, that $f(x)$ is an even function and that $g(x)$ is an odd function. Would $a^{f(x)}$ be an even or odd function? And would $a^{g(x)}$ be an even or odd ...
0
votes
2answers
21 views

Subadditive sequence implies $\lim s_n/n$ exists.

How do I prove that for a subadditive sequence the next limit $$\lim s_n/n$$ exists where $s_n$ is subadditive? PS I can see that $s_n/n$ is subadditive as well, but I don't see how to use it here.
0
votes
1answer
32 views

Convergence of a non explicit sequence

I was wondering if someone could help me prove this: Let $\lbrace a_n \rbrace$ be a sequence in $\mathbb{R}$ with ${a_n} \rightarrow a $ as $n \rightarrow \infty$. If $a>0$ prove that there exists ...
1
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0answers
19 views

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!
0
votes
2answers
53 views

The product of uniformly continuous functions is 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 ...
1
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1answer
40 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, ...
-1
votes
2answers
31 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 ...
1
vote
1answer
26 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} } = ...
1
vote
2answers
49 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 $-\infty<x_1,x_2<\infty$, ...
0
votes
1answer
24 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 ...
0
votes
0answers
41 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
vote
1answer
16 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 ...
4
votes
3answers
41 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 ...
1
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0answers
21 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
votes
1answer
30 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
votes
1answer
23 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 ...
1
vote
0answers
31 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 ...
3
votes
1answer
32 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 ...
-1
votes
2answers
118 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 ...
0
votes
0answers
47 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
votes
1answer
41 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 ...
0
votes
1answer
37 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
votes
2answers
50 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
votes
1answer
22 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} ...
-4
votes
2answers
56 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 ...
1
vote
2answers
52 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.
0
votes
1answer
20 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
vote
1answer
45 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
votes
1answer
36 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 ...
0
votes
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
votes
1answer
21 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 ...
1
vote
0answers
12 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 $n$ ...
1
vote
2answers
45 views

$L^{p}$ spaces and their properties

I have aquestion :Idont know how to show that if $1<p<q<\infty$ , then $L^{q}$(0,1)$\subset$$L^{p}$(0,1) and $\mid\mid f\mid\mid$$_p$ < $\mid\mid f\mid\mid$$_q$ ,f $\in$$L^{q}$(0,1)? ...
3
votes
2answers
38 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
votes
1answer
39 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
votes
2answers
24 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 ...
0
votes
0answers
22 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
votes
1answer
27 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 ...
0
votes
0answers
28 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
vote
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 ...
-1
votes
0answers
11 views

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 ...
1
vote
1answer
36 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
votes
1answer
24 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 ...
1
vote
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.
0
votes
1answer
22 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 ...
0
votes
1answer
12 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, ...
0
votes
1answer
11 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 ...