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...
2
votes
0answers
9 views
differentiability of the inverse function of $b^x$
I want to calculate the derivative of $\log_b(x):\mathbb R^+\rightarrow\mathbb R$ for $b>0,b\neq1$.
I want to use the following theorem from lectures:
Let $f:[a,b]\rightarrow\mathbb R$ be ...
1
vote
0answers
21 views
Prove that the dimension of the tangent space $T_x(X)$ of a k-dimensional manifold is k
In section 2, page 9 of Guillemin and Pollack's book $\textit{Differential Topology}$, he gave a proof that the dimension of the tangent space $T_x(X)$ is equal to the dimension of the manifold $X$. ...
0
votes
1answer
33 views
Continuous function on a closed set
Let $f: F \to \mathbb R$ be defined in a closed set $F \subset \mathbb R$. Show that $f$ is continuous if and only if for all $c \in \mathbb R$, the sets $E[f \le c]=\{x \in F; f(x) \le c\}$ and $E[f ...
0
votes
0answers
31 views
Simple approximation in $L^\infty$ is not possible.
I know that simple approximation in $L^p$ for $1 \le p \lt \infty$: For any $f \in L^p$ and $\epsilon >0$, there is a simple function $\phi \in L^p$ such that $||f-\phi||_p<\epsilon$.
But why ...
0
votes
1answer
25 views
Calculation of a multivariable integral
I read a paper which contains a tedious calculation and end up with the following integral:
$$
\int_{\mathbb{R^2}}\int_{\mathbb{R^2}}\frac{1}{1+\|x\|^2}\frac{1}{1+\|x+y\|^2}\frac{1}{1+\|y\|^2} \, ...
1
vote
0answers
16 views
Indirect Concavity
Let g: $\mathbb{R}^2 \rightarrow \mathbb{R}$ be defined by g(x) = $e^{min{(x1,x2)}}$ at each $ x\in \mathbb{R}^2$. Find whether or not g is indirect concave on $\mathbb{R}$.
0
votes
0answers
16 views
$\mathrm{E}[\log (1 + a X)]$ for non-central chi-squared distributed $X$
I'm really not great in analysis, so I recently got stuck on this problem (Please correct me if I'm wrong somewhere):
Let $Y \sim \mathcal{N}(\mu, 1)$ be a random variable with mean $\mu$ and normal ...
1
vote
2answers
28 views
Question on Contractions
Let $S = \{x \in \mathbb{R}^n ; \|x\| \le 1 \}$ and $f: S \to S$ be a contraction. Determine one can have $f(S) = S$.
I really need some help with this question. In advance I wanted to give all ...
3
votes
1answer
22 views
Any arbitrary closed smooth curve bounds a orientable surface?
I've got a question that, given an arbitrary closed smooth curve $C:[0,1]\rightarrow\mathbb{R}^3$, can you always find a orientable surface $\Omega$ which satisfy $\partial\Omega=C[0,1]$ ?
I have no ...
2
votes
1answer
38 views
Uniform convergence and uniform boundedness
I try to understand a demonstration from a book, but I have a problem with a line.
We have the series
$$u(x,t) = \sum_{k=0}^\infty \frac{g^{(k)}(t)}{(2k)!}x^{2k} \qquad (*)$$
where
$$g(t) = \left\{ ...
2
votes
1answer
35 views
Finding the maximum and minimum of $f$ on a set $Q$
I stumbled across a proposed task that I'm unable to solve.
We have a function $f:\mathbb{R^2}\to\mathbb{R}$ defined as:
$$
f(x,y):=x^2+xy+y^2+x+y+1
$$
The task is to "find the maximum and minimum" ...
1
vote
0answers
21 views
Example on Correspondences
Give an example of correspondences F: X $\rightarrow$ Y, G: Y $\rightarrow \mathbb{R}^s$ such that F and G are closed, but (G o F) is not, if any, where $ \varnothing \neq X \subset\mathbb{R}^m, ...
6
votes
4answers
61 views
Open Cover / Real Analysis [duplicate]
I have the next question: Let $K \subset $ $R^1$ consist of $0$ and the numbers 1/$n$, for $n=1,2,3,\ldots$ Prove that $K$ is compact directly from the definition (without using Heine-Borel).
I'm ...
0
votes
1answer
29 views
Multiplication in $\mathcal D'(R)$.
I was reading in my text book that multiplication of elements of $\mathcal D'(R)$ is not a continuous function although it is defined and can be seen from the convergence of $\sin(nx)$ to $0$ as $n$ ...
2
votes
2answers
39 views
which of the followings are true for bijective functions
which of the followings are true:-
1. There is a continuous bijection from $\mathbb{R}^2\to \mathbb{R}$.
2. There is a bijection between $\Bbb{Q}$ and $\mathbb{Q}\times \mathbb{Q}$.
Can somebody ...
0
votes
2answers
119 views
Prove that $\int_{-\infty}^{\infty} \sin x \, dx = 0 $
$$\int_{-\infty}^{\infty} \sin x \, dx$$
When I am doing the proof for this, why do i have to split it into
$\int_{-\infty}^a \sin x \, dx + \int_a^\infty \sin x \, dx $?
where a is a constant
1
vote
0answers
41 views
Antiderivative of an absolute function
$sgn(x)$ is the Sign-Function, $F$ is an antiderivative of $f$ and $S(x) := F(x) \cdot sgn(f(x))$
$$ \int \left|f(x)\right| \, dx = S(x) + \left(\sum\limits_{p=1}^{q}sgn(x-z_p) \lim_{x \to ...
1
vote
1answer
47 views
How to know a function is integrable or not?
Let say
$$ h(x) = \begin{cases}x^2,& x \in \mathbb {Q}\\-x^2,& x \notin \mathbb{Q}\end{cases}
$$
Is there a difference between riemann integrable and integrable?
And can I just ...
3
votes
2answers
39 views
Rudin Theorem 1.17
1.17 THEOREM
Let $f: X \to [0, \infty)$ be measurable. There exist simple measurable functions $s_n$ on $X$ such that
(a) $0 \le s_1 \le s_2 \le \dots \le f.$
(b) $s_n(x) \to f(x)$ as $n \to \infty ...
2
votes
1answer
30 views
Every subsequence of $x_n$ has a further subsequence which converges to $x$.Then the sequence $x_n$ converges to $x$.
Is the following is true?
Let $x_n$ be a sequence with the following property: Every subsequence of $xn$ has a
further subsequence which converges to $x$. Then the sequence $x_n$ converges to $x$.
I ...
3
votes
2answers
52 views
Searching for unbounded, non-negative function $f(x)$ with roots $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$
If a function $y = f(x)$ is unbounded and non-negative for all real $x$,
then is it possible that it can have roots $x_n$ such that $x_{n}\rightarrow \infty$ as $n \rightarrow \infty$.
0
votes
0answers
18 views
Helly's selection theorem
Can someone guide me to a reference (preferably open access online) stating and proving Helly's selection theorem for sequences monotone uniformly bounded functions on $[0,1]$. Something that can ...
0
votes
0answers
21 views
If a function $f:J\to\mathbb{R}$ satisfies the Zygmund condition, is it $C^1$?
A function $f\colon J\rightarrow \mathbb{R}$ on an open interval $J$ satisfies Zygmund condition if, for all
$x,y\in J$, $$f(x)+f(y)-2f\left(\frac{x+y}{2}\right)=o(|x-y|).$$ It is clear, if $f\in ...
1
vote
1answer
68 views
Is every convergent sequence Cauchy?
Wikipedia: "Every convergent sequence (with limit s, say) is a Cauchy sequence, since, given any real number ε > 0, beyond some fixed point, every term of sequence is within distance ε/2 of s, so any ...
1
vote
2answers
35 views
How can i show this inequality?
Let $n>1$ and $a_1,...,a_n \in \mathbb{R}^+$ be such that $\sum a_i=1$. For evey $i$, define $b_i=\sum_{j=1,j\neq i}a_j$. Show that
$\sum_{k=1}^n \dfrac{a_k}{1+b_k}\ge \dfrac{n}{2n-1}$
Thanks a ...
1
vote
1answer
26 views
Is this set relatively open or closed?
Take the set $\{(x,y) : (x-2)^2 + y^2 < 2\}$ . Is this set relatively open or relatively closed in the subspace $B(2,0)$ radius $\sqrt2$. (The open ball centered at $(2,0)$ with radius $\sqrt2$) ...
2
votes
1answer
31 views
limsup and liminf and the product of sequences
I'm trying to show that if $ \limsup s_n = +\infty$ and $\liminf t_n > 0$, then $\lim\sup s_n t_n = +\infty$.
Could someone check my proof/give feedback?
Since $\lim\inf t_n > 0$, we know that ...
1
vote
1answer
24 views
Prove that $d_2=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$.
I am trying to prove that $d_2\left({x,y}\right)=\sum_{j=1}^{k}\left|{x_j-y_j}\right|$ is a complete metric on $\mathbb{R}^k$. Here is my reasoning for why it is, but I am unsure about its ...
-1
votes
2answers
73 views
If $f: \mathbb{C}\to\mathbb{C}$ is bounded, then is it a constant? [closed]
If a function $f: \mathbb{C}\to\mathbb{C}$ is bounded, then it is a constant. Is it true or false?
2
votes
1answer
36 views
Weierstrass $M$-test problem, $f_n(x)=(nx^2)/(n^3+x^3)$
Use the Weierstrass M-test to show $$f(x)=\sum_{n=1}^\infty \frac{nx^2}{n^3+x^3}$$ converges uniformly on any finite interval $[-R,R]$.
This was an exam question I had. My attempt was to find an ...
0
votes
0answers
46 views
Can anyone see why this lemma is true? Seems very confusing!
Given a function f, we write $\bar{f}(t) = sup_{u \leq t} f(u).$
Lemma: Let $s_{0},.....,s_{T}$ be real numbers and $h:\Re \longrightarrow \Re.$ Then
$$\displaystyle ...
4
votes
4answers
125 views
How is the boundary of the clopen set [0,1) empty?
I don't get why the boundary of a clopen set is empty. If you take A = [0,1) in R, then isn't the closure of this the smallest closed super set that contains A..which is [0,1]. Isn't the interior, ...
4
votes
0answers
75 views
Is there a subsequence of $a_n = n \sin(n)$ which tends to $0$?
I know there is such a subsequence for $b_n = \sin(n)$. What about $a_n = n\sin(n)$?
0
votes
1answer
33 views
Riemann integral show $f(x)=g(x)$ for at least 1 $x$ in [a,b]
Let $f$ and $g$ be continuous functions on $[a,b]$ such that $\int_a^b f = \int_a^b g$. Show that there exists $x\in [a,b]$ such that $f(x) = g(x) $.
I want to assume not and then show that the ...
1
vote
1answer
32 views
Monotonic integral proof
Let $f$ be a continuous function on $[a,b]$ such that $f(x) \geq 0 $
for every $x\in [a,b]$. Suppose $\int_a^b f = 0$ and show that $f (x) = 0$ for every $x\in [a,b]$.
obv this is monotonic ( ...
1
vote
1answer
41 views
Composition of Lebesgue measurable function $f$, with a continuous function $g$ having a certain property, is Lebesgue measurable
Suppose that $f$ is Lebesgue measurable and $g$ is real valued, continuous, and has the property that for any null set $N$, $g^{-1} (N)$ is measurable. Then $f \circ g$ is also Lebesgue measurable. ...
0
votes
0answers
24 views
conditions on coefficients of univariate polynomial so that it has only real roots
Consider a univariate polynomial of degree $n$ with real coefficients. Are there general equalities/inequalities on its coefficients, so that it has precisely $n$ real roots?
For example for the case ...
0
votes
1answer
37 views
space of riemann integrable functions not complete
Define norm as $\int |f|$ (Riemann integral) on $\mathcal R^1[0,1]$, the space of riemann integrable functions on $[0,1]$ with identification $f=g$ iff $\int |f-g|=0$.
Let $\{ r_1,r_2,\cdots \}$ be ...
0
votes
1answer
44 views
Show $C\geq \mathrm{max}\left \{ A,B \right \}$.
Let $\sum_{n=0}^{\infty}a_{n}x^{n}$ and $\sum_{n=0}^{\infty}b_{n}x^{n}$ be the power series with the convergent of radius respectively $A>0$ and $B>0$.
Define $c_{n}=\mathrm{min}\left \{ ...
0
votes
0answers
42 views
Uniform convergence of $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$.
Given that the series $\sum_{n=2}^{\infty}\dfrac{\mathrm{ln}(n)^{p}}{n}$ is convergent iff $p<-1$.
Show that $\sum_{n=2}^{\infty}\frac{\mathrm{ln}(n)^{x}}{n}$ is uniform convergent as $x \in ...
0
votes
1answer
40 views
Find for all value of constant $a>0$; the interval of convergence of the power series $\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$
Find for all value of constant $a>0$; the interval of convergence of the power series
$\sum_{n=0}^{\infty}\frac{1}{1+a^{n}}x^{n}$.
What I have tried is; if we let $b_{n}=\frac{1}{1+a^{n}}x^{n}$ so ...
1
vote
2answers
61 views
If $K$ is compact, then $C(K,\mathbb{R}^n)$ is a Banach space under the norm $\|f\|=\sup_{x\in K} \|f(x)\|$
Let $K$ be a topological space that is compact. Show that the space $C(K,\mathbb{R}^n)$ of all the continuous functions $f:K\to\mathbb{R}^n$ is a Banach space with the norm $\|f\|=\sup_{x\in K} ...
1
vote
4answers
38 views
Is $(a_n)_n$ with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ a Cauchy sequence?
Let $0 < q < 1$ and $(a_n)_n$ be a squence with $|a_{n+2}-a_{n+1}| ≤ q|a_{n+1}-a_n|$ for all $n ∈ ℕ$.
I need to show that this is a Cauchy sequence. I'm not sure how to start this one, as we ...
2
votes
0answers
26 views
Growth of partial sums of a divergent series
I am trying to find how the product $\prod_{n=0}^Me_n$ grows with $M$, when $$e_n=1+\frac{a_1}{t+n}+\frac{a_2}{(t+n)^2}+\dots$$
with large $t$. So as a first estimate I took $e_n=1+\frac{a_1}{t+n}$ so
...
0
votes
1answer
31 views
Show convergence for this sequence only by using the definition
I need to prove convergence for
$(b_n)_{n ∈ ℕ}=\left(\frac{(-1)^nn}{2n+1}\right)_{n∈ℕ}$ and also show the limit.
I may only use the following definition: $∀ɛ > 0∃n_0∈ℕ∀n≥n_0:|a_n-a|< ɛ$.
So far ...
10
votes
1answer
116 views
Uniformly convergence?
I'm not sure wether or not the following sum uniformly converge on $\mathbb{R}$ :
$$\sum_{n=1}^{\infty} \frac{\sin(n x) \sin(n^2 x)}{n+x^2}$$
Can someone help me with it? (I can't use Dirichlet' ...
0
votes
1answer
18 views
Show relation for integrals
Let $f \in C^{1}([a,b];\mathbb{R})$ and $|f'(x)-f'(y)| \le L |x-y|$
then we have $|\int_a^b f(x) dx -f(\frac{a+b}{2})(b-a)| \le L\frac{(b-a)^3}{4}$.
I have troubles to show this inequality. the ...
1
vote
1answer
49 views
A question about “nice” functions
Let $f:\mathbb R \rightarrow \mathbb R$ be a function such that $\lambda(I)=\lambda(f(I))$ for each interval $I \subseteq \mathbb R$. ($\lambda$ is Lebesgue measure here.) Let us call such functions ...
1
vote
2answers
40 views
Proving that Euclidean space having the infinity metric is a complete metric space (stuck)
I am trying to prove that the space ${\mathbb{R}}^k$ with the $\infty$-metric is a complete metric space.
I know that I need to show that every Cauchy sequence in the metric space ${\mathbb{R}}^k$ ...
0
votes
1answer
96 views
Changing order of derivatives
I would like to rewrite the following expression
$$\frac{d^i}{dx^i}\left\{f(x)\left[\frac{d^jf(x)}{dx^j}\right]\left[\frac{d^kf(x)}{dx^k}\right]\right\}$$
into the form
$$D f(x)^3,$$
with $D$ ...
