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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0
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22 views

Test for absolute convergence $\sum_{k=1}^\infty \frac{(-1)^{k+1}k^k}{(k+1)^k}$

Test for absolute and conditional convergence. $$\sum_{k=1}^\infty \frac{(-1)^{k+1}k^k}{(k+1)^k}$$ $$\lim_{k\to\infty}|a_n| = \lim_{k\to\infty} \frac{k^k}{(k+1)^k}$$ I'm stuck on what to do next.
2
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0answers
35 views

Show that $\int_E f (x, y) dx$ is differentiable with respect to $y$ and $\frac{d}{dy}\int_E f(x,y)dx=\int_E \frac{d}{dy}f(x,y)dx.$

Assume that $f = f(x,y)$ is a function defined on $E × (a,b).$ For each fixed $y ∈ (a,b),$ $f$ is integrable with respect to $x$ on $E$, and for each fixed $x ∈ E$, $f$ is differentiable with respect ...
0
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3answers
57 views

Proof that the continuous image of a compact set is compact [duplicate]

Let $X\subset \mathbb R^{n}$ be a compact set, and $f :\mathbb R^{n}\to \mathbb R $ a continuous function. Then, $F(K)$ is a compact set. See, I know that this question may be a duplicate, but the ...
8
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2answers
206 views

A Binet-like integral $\int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{x^s }{1-x}\mathrm{d}x$

I met this integral $$ \int_{0}^{1} \left(\frac{1}{\ln x} + \frac{1}{1-x} -\frac{1}{2} \right) \frac{ \mathrm{d}x}{1-x} \qquad (*) $$ while evaluating this log-cosine integral. I made several ...
1
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0answers
20 views

Showing a function is upper semicontinuous

Let $f: \mathbb{R} \rightarrow [0, B]$ and for every $\varepsilon > 0$, let $\varphi_{\varepsilon}(x) := \sup_{\{y: |x - y| < \varepsilon\}}f(y)$. Since for each fixed $x$, ...
1
vote
1answer
31 views

Proving and visualizing $\mathbf 1_{(x,x+a]}(y) = \mathbf 1_{[y-a,y)}(x)$

Here is a trick from one of the proofs in probability: $$\iint \mathbf 1_{(x,x+a]}(y) \ \lambda(dx) \ \mathbb P(dy) = \iint \mathbf 1_{[y-a,y)}(x) \ \lambda(dx) \ \mathbb P(dy)$$ for $a>0$. So ...
3
votes
1answer
35 views

Calculating limit in parts. Why possible?

Let $f$, continuous function, differentiable at $x=1$ and $f(1)>0$. Consider the following equation: $$\lim \limits_{x\to 1} ...
3
votes
1answer
122 views

Monotone Convergence Theorem for Riemann Integrable functions

I'm having a really hard time proving this statement (this is not homework): If $f_{n} : [0,1] \rightarrow \mathbb{R}$ is a Riemann integrable function for all $n \in \mathbb{N}$, and $0 \leq f_{n + ...
2
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1answer
44 views

Are these two expression square integrable?

I have two expressions (let's call them functions $f,g$) on $[0,1]$, where I want to find out whether they are square-integrable or better: for which $m \in \mathbb{Z}$ they are square-integrable ( ...
0
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0answers
31 views

Convergence of norms

I have this space $H_{0,p}^1=\lbrace u\in AC([0,+\infty),\mathbb{R}),u(0)=u(+\infty)=0, \sqrt{p} u'\in L^2(0,+\infty)\rbrace $ endowed with the norm $||u||^2=\int_0^{+\infty} p(t) u'^2(t) dt$ ...
1
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1answer
36 views

Reverse Fatou's lemma on probability space

Let $(\Omega, \mathcal{F},\mathbb{P})$ be probability space and $E_{n \in \mathbb{N}}$ be $\mathcal{F}$-measurable sets. Show example that reverse Fatou's Lemma, $\mathbb{P}(\limsup_n E_n) \geq ...
2
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2answers
77 views

Pick a smart function

Our teacher wants us to find a function $f$ on $(0,\pi)$ such that $$\sqrt{\sin(x)} f(x)^{\frac{1}{4}} =k_1 + \cos(x)$$ and $$\sqrt{\sin(x)} f(x)^{-\frac{1}{4}} = k_2 + \cos(x).$$ The two constants ...
3
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0answers
57 views
+50

What types of integrals cannot be solved using improper Riemann-Stieltjes Integration?

I came across the wikipedia discussion of the Riemann-Stieltjes integral. The first sentence in the "Generalization" section gave me pause: An important generalization is the Lebesgue–Stieltjes ...
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0answers
14 views

Change of variables formula with integrator of bounded variation

Let $G$ be be continuous with bounded variation on finite intervals. If $f$ is continuous then it is well known that $\int_a^bf(G(s))dG(s)=\int_{G(a)}^{G(b)}f(x)dx$. How general can $f$ be so that ...
1
vote
1answer
39 views

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\cap_k B_k$ is either a point or a closed ball.

$B_k$ sequence of closed balls in $\mathbb{R}^n$ such that $B_{k+1}\subset B_k$, show $\bigcap_k B_k$ is either a point or a closed ball. Please help me check the proof, thanks! Define $x_k$ to be ...
0
votes
1answer
23 views

Understanding Distributional Meanings and Test Functions for PDEs

thank you for taking the time to read my question. My question is about distributional meanings in PDEs. My specific question is at the bottom, but I'd be interested in a bit of general theory (even ...
2
votes
4answers
51 views

Prove a sequence does not tend to zero

Prove that the sequence $\dfrac{a^n}{2^nn^2}$ where $a>2$ does not tend to zero. I thought about writing $a=2+{\epsilon}$ then using binomial expansion which is valid for ${\epsilon}<1$ but ...
0
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1answer
26 views

Find the limit $\lim\limits_{n\rightarrow +\infty} n^2 \int_0^{2n} e^{-n|x-n|}\log\left[1+\frac{1}{x+1}\right] dx$

Find the limit $$\lim_{n\rightarrow +\infty} n^2 \int_0^{2n} e^{-n|x-n|}\log\bigg[1+\frac{1}{x+1}\bigg] dx$$ I tried to change the variable $y=n(x-n)$, then we get $$\lim_{n\rightarrow +\infty} n ...
0
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1answer
48 views

Prove that $g$ is continuous. [duplicate]

Let $f:[a,b]\to \mathbb{R}$ be a continuous function and let $g:[a,b]\to \mathbb{R}$ be a function such that $g(a)=f(a)$ and $g(x)=\sup_{t\in [a,x]}f(t)$. Prove that $g$ is continuous on $[a,b]$. ...
2
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1answer
18 views

Hadamard's Lemma in multidimensional real analysis

This is Hadamard's Lemma: Let $U \subset \Bbb R^n$ be an open set, let $a \in U$ and $f: U \to \Bbb R^p$. Then the following assertions are equivalent. The mapping $f$ is differentiable at $a$. ...
1
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1answer
47 views

Subsequences and limit inferior

Suppose $f\colon \mathbb{R} \to \mathbb{R}$ is a continuous function. Let $x\in \mathbb{R}$ and let $(x_n) \subseteq \mathbb{R}$ be a sequence converging to $x$. Let $(y_n)$ be a subsequence of ...
2
votes
1answer
24 views

convolution of compactly supported continuous function with schwartz class function is again a Schwartz class function?

Suppose $f$ is continuous function on $\mathbb R$ with compact support; and $g\in \mathcal{S}(\mathbb R),$ (Schwartz space) My Question is: Can we expect $f\ast g \in \mathcal{S(\mathbb R)}$ ? ...
1
vote
1answer
106 views

Show there exists $x\in (0,1)$ such that $f(x) \leq \int_0^1 f(t) dt$

Please help me check my proof, thanks! (a) Show there exists $x\in (0,1)$ such that $$f(x) \leq \int_0^1 f(t) dt.$$ Proof: when $f$ is constant a.e, the equality holds for all points except for a ...
0
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1answer
17 views

$f\in L^{p}(\mathbb R)\cap C_{0}(\mathbb R); (1<p<\infty), g\in C^{\infty}_{c}(\mathbb R) \implies f\ast g \in C^{k}(\mathbb R)$?

We put, $C_{0}(\mathbb R)=$ The space of continuous functions on $\mathbb R$ vanishing at $\infty$; $C^{k}(\mathbb R)=$ The space of all functions $\mathbb R$ whose derivative of order $\leq k$ exist ...
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0answers
17 views

Cardinality of the following set of functions on $\mathbb R$ [duplicate]

Consider the following set $W$ = The set of constant functins on $\mathbb R$. $X$ = The set of polynomial functins on $\mathbb R$. $Y$ = The set of continous functins on $\mathbb R$. $Z$ = The set ...
1
vote
1answer
33 views

Proof that this set is not compact

Let $X=C[0,1]$ with the $\sup$ norm. Let $Y = \{f\in X\mid \|f\|_\infty \le 1\}$. It is my goal to show that $Y$ is not compact using the sequence defintion of compactness. Note that it is very easy ...
2
votes
2answers
102 views

How to reduce this to Sturm-Liouville form?

I have the ODE $$-(1-x^2) \frac{d^2 f(x)}{dx^2} + x \frac{df(x)}{dx}+g(x)f(x)=\lambda f(x)$$ and I want to reduce it to Sturm-liouville form. The problem is that we don't have $2x$ but just $x$. ...
4
votes
3answers
109 views

Prove $ne^{-n}$ converges to zero

How would I prove that $ne^{-n}$ converges to zero? I've tried $ne^{-n}<{\epsilon}$ and then logging both sides but no further progress could be made. Thanks
1
vote
1answer
25 views

Equivalence of Localized Fourier Restriction Estimates

I'm reading Tao's Park City notes on the restriction conjecture http://arxiv.org/abs/math/0311181. He says at some point that the estimate: there is a constant $C > 0$ such that, for any $R \ge ...
2
votes
1answer
32 views

Does the limit $\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$ always exist?

Let $f$ be a Lebesgue integable function. Does the limit $$\lim_{k\to\infty}\int|\cos kx |f(x) d\lambda(x)$$ always exist?
2
votes
1answer
50 views

There exist $c(c\in \left(\frac{1}{2},1\right)$ s.t $2 \int _{0}^{1}f(x)dx=\frac{f(c)}{c} $

I would appreciate if somebody could help me with the following problem: Let $f(0)=0,f(1)=1,f'(x)>0,f''(x)<0$ on $[0,1]$ Q: Under the proposition is true of false? There exist $c(c\in ...
1
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0answers
15 views

existence of solution of volterra integral equation of the first kind

$$ \int_0^t k(s,t)f(s)ds=g(t) $$ To know the existence and uniquness of solution of volterra integral equation(VIE) of the first kind, we differentiate it and convert to the VIE of the second kind. ...
0
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1answer
52 views

Doubt on an ODE problem

Consider the following differential equation $$x'(t) = h(x(t))$$ Consider a function $x(t)$ which satisfies the differential equation for $0 \lt t \leq 1$ and another function $y(t)$ for $0.5 \leq ...
3
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1answer
24 views

Prove this function is absolute continuous and Lipschitz of order $\alpha$

Let $1≤p≤\infty$ and $ f \in L^{p}(a,b)$ such as there is a function $g \in L^{p}(a,b)$ that for all $\phi \in C^{1}(a,b)$ (and continuos in $[a,b]$) with $\phi(a)=\phi(b)=0$ we have: $\int_{a}^{b} ...
-1
votes
1answer
50 views

If $f$ is in $R[a,b]$, show that [on hold]

If $f$ is in $R[a,b]$, show that $$\int_a^b f =\lim_{c\to a^+} \int_c^b f$$ the hint is to show $$\left|\int_a^b f - \int_c^b f \right| = \left|\int_a^c f \right| \le \int_a^c |f|$$
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2answers
26 views

radius of convergence when root test fails

I'm stuck on this problem: Find the radius of convergence of $\sum \limits_{n=1}^\infty \frac{x^n}{((2+(-1)^n)^n} $ An attempt: From the root test, it seems $L$ does not exist: $$L= \lim_{n ...
0
votes
1answer
28 views

Cauchy Sequence of Differentiable Functions Implies Cauchy Sequence of Derivatives?

Can someone provide a proof of the fact that if I have a sequence of differentiable functions that is Cauchy and converges uniformly on some interval $[a,b]$, then the sequence of their derivatives ...
2
votes
1answer
49 views

Proof: $(\sup(A) - \epsilon)^n<y<(\sup(A)+\epsilon)^n$

Prop.: let be $y \in \Bbb{R}_{>0}$, $n \in \Bbb{N}_{>0}$, and $A \subseteq \Bbb{R}$, then: $$A=\{x| x \in \Bbb{R}_{>0}\wedge x^n \leq y \} \Rightarrow (\sup(A) - \epsilon)^n< ...
0
votes
2answers
48 views

Define $f:\mathbb{R}\rightarrow\mathbb{R}$ by $f(x)=x-[x]$. Determine those points at which $f$ has a limit, and justify your conclusions.

$[x]$ denotes the largest integer that is less than or equal to $x$. Ok, so I can see that $f$ has a limit at every point in $\mathbb{R}\setminus\mathbb{Z}$, but I am having a difficult time ...
2
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4answers
113 views

Proving exponential is growing faster than polynomial

Let $P(x)$, a polynomial which isn't the zero-polynomial. I want to prove the following limits $$\lim \limits_{x\to\infty} \left|P(x)\right|e^{-x} = 0$$ $$\lim \limits_{x\to-\infty} ...
1
vote
1answer
21 views

Continuity of a function defined by an integral, when the variable is in the region of integration

Hi everyone: Suppose that $f$ is locally integrable in $\mathbb{R}^{n}$, $(n\geq2)$; $B(a,r)$ is the ball of center $a$ and radius $r>0$ , and $\lambda$ is the $n$-dimensional Lebesgue measure. It ...
1
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0answers
17 views

Integrating with indicator functions

I want to evaluate $$\int_{-\infty}^{\infty}(A_1e^{-\beta_1(b-x-y)}+B_1e^{-\beta_2(b-x-y)})(pn_1e^{-n_1y}1_{\{y\geq0\}}+qn_2e^{n_2y}1_{\{y<0\}})dy,$$ $b>x, \beta_1<n<\beta_2$. I am trying ...
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0answers
6 views

existence of symmetrization algorithm

For symmetrization of a Borel set, the construction of an an explicit algorithm (i.e. a sequence of symmetrization steps that will lead to a ball) is an open question. But have we proved the ...
2
votes
2answers
70 views

The trace of an operator

My question is derived from A. Deitmar's book: A First Course in Harmonic Analysis (second edition), p22, Exercise 1.17. Let me rewrite it again: Let $k:\mathbb{R}^2 \rightarrow \mathbb{C}$ be smooth ...
5
votes
1answer
64 views

Visualisation of the smash product

wedge product, join etc. all of them are no problem for my head, but I am really failing to get a visual idea of what the smash product wants to tell me. For example if I take two spheres, I have no ...
6
votes
1answer
49 views

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set.

$\int_0^1 (f(x))^n =$ constant, $f\geq 0$, then $f$ is a characteristic function of a measurable set. This is the result from question part (a). Now for part (b), will it also hold when the ...
1
vote
3answers
38 views

Dense subsets of $(L^p(\Omega),\|\cdot\|_p)$

The following results hold. Theorem Let $\Omega\subset\mathbb{R}^n$ be an open set. Then $C^0_c(\Omega)$ is dense in $(L^p(\Omega),\|\cdot\|_p)$, if $1\le p<\infty$. Theorem Let ...
4
votes
2answers
53 views

Borel measure supported on $\mathbb{Q}$

Let $\mu$ be a Borel measure supported on $\mathbb{Q} \subset \mathbb{R}$. Must $\mu$ be a sum of Dirac measures?
1
vote
1answer
49 views

$C^0$ is a closed subspace of $L^{\infty}$

Let $\Omega\subset\mathbb{R}^n$ be an open bounded set. Let $f\in C^0(\bar\Omega)$. I have to prove that $\|f\|_{\infty}=\|f\|_{L^{\infty}}$. One implication is trivial. Let's consider the other one. ...
2
votes
4answers
136 views

How to finish this proof about compact implies bounded

A set is called compact if every sequence has a convergent subsequence. I am trying to show: If $K$ is compact then it is bounded. (that it is closed was very easy to prove) What I want to do: Let ...