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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10 views

Why does $y(s)$ continuous imply that $f(s)$ with $f_l (s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum \max\{0,z_l(s)\}}$ is continuous?

Let $z:\triangle^{L-1}\to \mathbb{R}^L$ be continuous. Define $f:\triangle^{L-1} \to \triangle^{L-1}$ be defined component wise as $$ f_l(s) = \frac{s_l + \max\{0,z_l(s)\}}{1+\sum_{l=1}^L ...
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1answer
35 views

existence of neighbor of linear transformation T such that…

I want to show if a linear transformation $T$ has rank $k$ then there exist $\delta$ such that the open ball centered at $T$ with radius $\delta$ contains only linear transformations that has rank ...
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2answers
51 views

Two disjoint closed sets $A, B \subset \mathbb{R}$ such that there does not exist $ϵ$ with $d(A, B)> ϵ$

Show that there exist two disjoint closed sets $A, B \subset \mathbb{R}$ such that there does not exist a positive $ϵ$ with $d(x, y)\ge ϵ$ where $x∈A$ and $y∈B$. I have already proved that if $A$ ...
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1answer
35 views

Show that there exists c such that $f(c)=c^2$ [on hold]

Let $f:[0,1] \rightarrow [0,1]$ be a continuous function. Show that there exists c in $[0,1]$ such that $f(c)=c^2$
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25 views

Well-ordering principle proof via Analysis

I would appreciate if my proof attempt could be evaluated, and some hints could be given. I think that, perhaps, my proof is not ideal. Prove: If $E$ is a non-empty subset of $\mathbb{N}$ then $E$ ...
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3answers
57 views

What are the lightest hypothesis needed to be able to get the limit inside the integral?

Let $\{f_n\}$ be a sequence of Riemann integrable functions. What are the lightest conditions on $f_n$ to guarantee the following? $$ \lim_n \int_a^b f_n\,\mathrm dx=\int_a^b \lim_n f_n \, \mathrm ...
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1answer
15 views

Convergence of two sequences conjecture

If $(a_n) \rightarrow 0 $ and $(b_n - a_n) \rightarrow 0$, then $(b_n) \rightarrow a$. I can't think of any counterexamples to this conjecture but I'm not sure how to prove it. Any help would be ...
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2answers
47 views

Intersection of uncountable sets.

Let $C \subseteq [0,1]$ be uncountable. Let $A$ be the set of all $a \in (0, 1) $ such that $ C \cap [a,1] $ is uncountable and set $\alpha = \sup A$. Is $C \cap [\alpha, 1]$ an uncountable set? I ...
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2answers
51 views

If $f: \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a function with $\nabla f(x) = 0$ for all $x$, then $f$ is constant.

Suppose that $f$ is differentiable and that $\nabla f(x) = 0 \forall x \in \mathbb{R}^n$. Prove that there is a $v \in \mathbb{R}$ such that $f(x) = v \\ \forall x \in \mathbb{R^n}$ could anyone ...
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25 views

Prove or Disprove If $T : R^n \rightarrow\ell_2$ is linear, then $T(A)$ is totally bounded in $\ell_2$ when $A ⊆ R^n$ is bounded

Prove or Disprove, If $ T: \mathbb R^n \rightarrow \ell_2 $ is linear, then It preserves total boundedness $ T(A) $ is totally bounded in $\ell_2$ when $A \subseteq\mathbb R^n$ is bounded. I think ...
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4answers
85 views

Showing that if $\lim_{x\to\infty}f'(x)=L$ then $\lim_{x\to\infty}\frac{f(x)}{x} = L$.

Let $f:[0,\infty)\to\mathbb{R}$ derivable and suppose that $$\lim_{x\to\infty}f'(x)=L.$$ How can I prove that $$\lim_{x\to\infty}\frac{f(x)}{x} = L$$? I have solved some similar problems using the ...
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2answers
78 views

Suppose that $f(0)=f(2\pi)$. Show that there exists an x such that $f(x)=f(x+\pi)$.

I am supposed to show that there exists an $x$ in the interval $[0,\pi]$ such that $f(x)=f(x+\pi)$ by considering another function $g:[0,\pi] \to \mathbb{R}$ defined by $g(x)=f(x)-f(x+\pi)$. Should I ...
1
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1answer
13 views

Bump functions converging to an indicator

Suppose $K\subset\mathbb{R}^n$ has a smooth boundary, and let $\phi_s(x)$ be bump functions converging pointwise to the indicator of $K$, i.e. ...
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2answers
28 views

limit in integral?

Let $f$ be a smooth function : can someone tell me why we have : $\lim\limits_{\epsilon \rightarrow 0}\frac{1}{\epsilon}\int_{t}^{t+\epsilon}f(s)ds=f(t)$ thank you very much !
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1answer
27 views

What is an example of $f \in L^1(\mathbf{R})$ such that $\sum_{n=1}^\infty f(nx)$ converges a.e. but is not in $L^1(\mathbf{R})$

what is an example of $f \in L^1(\mathbf{R})$ such that $\sum_{n=1}^\infty f(nx)$ converges a.e. but is not in $L^1(\mathbf{R})$? Context: This question appeared on an old qualifying exam. I tried ...
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2answers
31 views

Further generalising Holder's inequality

I have proved the following theorem in an earlier part of the question: Let $p,q \geq 1$ be such that $\frac{1}{p} + \frac{1}{q} = 1$. Show that: $$\|fg\|_1 \leq \|f\|_p \|g\|_q$$. I proved this ...
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1answer
31 views

$\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$

I'd appreciate if you could please check to see if my proof is valid. Find $\sup$ and $\inf$ of $E=\{p/q\in\mathbb{Q}:p^2<5q^2 \text{ and } p, q > 0\}$. Solution: $q^2 > p^2/5 \iff q > ...
2
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1answer
16 views

Continuity on the parameters of the intermediate value theorem

Let $X$ be a compact metric space (feel free to impose more conditions as long as they're also satisfied by spheres) and $F : X \times [0, 1] \to \mathbb{R}$ a continuous function such that $F(x, 0) ...
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2answers
24 views

Question concerning limit superior and inferior.

Let ${a_n}$ and ${b_n}$ be two real sequences such that $a_n\leq b_n$ for all $n$. Is it true that $\lim \sup a_n \leq \lim\inf b_n $? Outline the proof if so.
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1answer
52 views

Show $A=\limsup_\limits{n\to\infty}a_n$.

Let $\{a_n\}$ be a sequence of real numbers bounded from above, $A\in \Bbb R$. Given any $\epsilon>0$, a)$\exists n_0 \in \Bbb N$ such that $ a_n<A+\epsilon$ for all $n\ge n_0$. b)$\exists ...
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2answers
19 views

If $Z(f)$ is the zero set, prove that $Z(f)$ is closed

Introduction: Exercise from Principles of Mathematical Analysis, third edition (Rudin), page 98. Exercise: Let $f$ be a continous real function on a metric space $X$. Let $Z(f)$ (the zero set of ...
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0answers
15 views

Integral inversion

Say I know this function $$ F(u) = \int _{-\infty}^{\infty}f(x) m\left(\frac{u}{x}\right) \mathrm d x$$ where $m(x)$ is a Fourier transform of an infinitely differentiable real function, whose maximal ...
4
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2answers
72 views

$f_n \geq 0$ and $\int f_n = 1$ implies $\limsup_n \left( f_n(x) \right)^{\frac{1}{n}} \leq 1$ for a.e. $x$

I am studying for a qualifying exam and am having difficulty with this problem: Let $\left( X, \mathcal{M}, \mu \right)$ be a measure space and assume $f_n \geq 0$ such that $\int f_n = 1$ for all ...
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2answers
31 views

Prove a function approaches infnity when the deriviative is greater than $0$

Here's my question: Let $f$ be a function which has a derivative in $\Bbb R$ such that $f'(x)\geq0$ and $f''(x)\geq0$ for all $x \in \Bbb R$ Prove that if there is some $a \in \Bbb R$ such ...
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2answers
38 views

Meaning of $\lim_{n \rightarrow \infty } \sup$ $ a_n$

Consider a real sequence $\{ a_n \}_{n \in \mathbf{N}}$. Now if I write $A = \lim_{n \rightarrow \infty }$ $ \sup$ $ a_n$ , what do I actually mean by this? I mean if $a_1 , a_2, . . . $ is my ...
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1answer
30 views

Disc of convergence involving logs

Find the disc of convergence: $$\sum_{n=2}^\infty \frac{z^{n}}{n(log(n))^p};(p>0)$$ I have tried geometic series, ratio test, root test... What would be your thought on the best test to use?
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2answers
131 views

Computation of an iterated integral

I want to prove $$\int\limits_{-\infty}^\infty\int\limits_{-\infty}^\infty\frac{\sin(x^2+y^2)}{x^2+y^2}dxdy=\frac{\pi^2}{2}.$$ Since the function $(x,y)\mapsto\sin(x^2+y^2)/(x^2+y^2)$ is not ...
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0answers
35 views

solution of linear elliptic equation

Can you please help me to show that if $\Omega\subset\mathbb R^n$ is a $C^2$ domain and $f$ is an application which belongs to $L^2(Ω)$ and $u$ is a weak solution of the linear elliptic equation: ...
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2answers
26 views

If $a_k\ge 0$ for all $k$ show that $\sum\limits_{k=0}^na_k\le b\in\mathbb{R}$.

Given that $\lim\limits_{x\rightarrow1^-} \sum\limits_{k=0}^\infty a_kx^k = b \in\mathbb{R}$ for $|x|<1$. If $a_k\ge 0$ for all $k$ show that $\sum\limits_{k=0}^na_k\le b$. This is just a step in ...
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0answers
25 views

On a max-min problem from an exam.

I have asked a different question on the same exercise (from an exam) a couple weeks ago, I hope it is acceptable to have a different question on the same exercise, I searched the Meta and it seems ...
1
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4answers
57 views

Show that a continuous function f either has a root hence $f(c)=0$ or $|f(x)|> e$ for $e>0$.

From what I understand, I am being asked to show that a function $f$ on an interval $[a,b]$ either has a root $c$ such that $f(c)=0$ or it does not have a root hence $|f(x)|\ge e$. Should I apply the ...
2
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1answer
39 views

Asymptotic behavior of a function

Let $n\geq 2$, and define $$\phi(t) = \int_{S^{n-1}} \cos(t \omega_1) d\omega,$$ where $S^{n-1}$ is the unit sphere in $n-$dimensional euclidean space, and $d\omega$ denote the surface area on ...
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1answer
26 views

How to calculate this limit with approximations?

$$\lim _{n\to \infty }\left(\frac{2^{\left(4n+1\right)}ln\left(2n^3+1\right)+n^5\cdot 10^n}{15^n+4^{\left(2n-1\right)}ln\left(5n\right)}\right)$$ I tried like that: $$\approx\lim _{x\to \infty ...
2
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0answers
35 views

How to prove this criteria of differentiability? [duplicate]

Let $f: I \to \mathbb{R}$ continuous and $a\in \operatorname{int}(I)$. Suppose that there is $L\in\mathbb{R}$ such that $$\lim \frac{f(y_n)-f(x_n)}{ y_n-x_n}=L$$ for all sequences $(x_n)$ and $(y_n)$ ...
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30 views

Dirac delta function

Is there any math book on which I can find a formal introduction to Dirac delta function?. Also which mathematical background would be needed to understand it? Thanks
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1answer
36 views

EDITTED: Find all values of $a$ and $b$ so that $ax^n+b\cos\left(\frac{x}{n}\right)$ is Cauchy.

For each $n\in\mathbb{N}$ let $$f_n(x)=ax^n+b\cos\left(\frac{x}{n}\right), \text{ } x\in[0,1].$$ Find all values of $a$ and $b$ for which $(f_n)$ is a Cauchy sequence in $C[0,1]$, the space of ...
0
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0answers
30 views

$\underset{x\rightarrow x_0}{\lim}f(x)=y_0$ iff $\underset{n\rightarrow \infty}{\lim}x_n=x_0$ implies $\underset{n\rightarrow \infty}{\lim}f(x_n)=y_0$ [duplicate]

I have the following task: Let $(X,d)$ and $(Y,e)$ be metric spaces, $E\subset X$ and $x_0$ be an accumulation point of $E$. We say that point $y_0\in Y$ is the limit point of mapping ...
2
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1answer
21 views

Measurability of set of numbers with infinite number of digits in decimal expansion equal to 8

Say A is the set of all Real numbers on $[0,1]$ whose decimal expansion contains an infinite number of 8s. I am trying to prove the measurability of this set. I realize that this is the set of ...
1
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1answer
25 views

Constant $a_{k}$ which normalizes integral is bounded

For $x \in [-\pi,\pi]$ define $$f_{k}(x) = a_{k}\cdot \left(\frac{1+\cos(x)}{2}\right)^{k}$$ where $a_{k}$'s are chosen such that $$\frac{1}{2\pi} \cdot \int_{-\pi}^{\pi} f_{k}(x) \ dx =1 $$ Then show ...
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4answers
71 views

If $\sum x_n$ converges absolutely . I'll have to show that $\sum \frac{x_n }{1+ x_n} $ converges . [duplicate]

I tried applying Dirichlet's test and Abel's test. Here I found some similar questions. But in most of the questions, they have assumed $x_n$ to be greater than 'zero' for all 'n', which allows them ...
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1answer
38 views

Prove that $\lim_n (-n+\sin(n) )= -\infty$

Prove that $\lim_n (-n+\sin(n)) = -\infty$ So I need to show that if $B'$ is any number, then there is a number N' such that $$n>N' \implies S_n\lt B'$$ I am having trouble feeling confident in ...
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3answers
69 views

Does the series converge?

I want to check if the series $$\sum_{n=2}^{\infty}\frac{1}{n\log n\log (n+1)}$$ converges. Let $a_n=\frac{1}{n\log n\log (n+1)}$. I found that $\frac{a_{n+1}}{a_n}\rightarrow 1$ so we cannot use ...
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4answers
45 views

How can we calculate the limit at $0$?

Suppose we have the function $g:\mathbb{R}\rightarrow \mathbb{R}$ and $g(x)=\left\{\begin{matrix} x\cdot (-1)^{[\frac{1}{x}]} & x\neq 0\\ 0 & x=0 \end{matrix}\right.$ I want to show that ...
1
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2answers
23 views

How do we show the limit?

We have that $(a_n)$ is a bounded sequence of real numbers that satisfy $2a_{n+1}\leq a_n+a_{n+2}, \forall n\in \mathbb{N}$. I want to show that the sequence $b_n=a_{n+1}-a_n$ converges and that the ...
0
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1answer
27 views

Equivalent Metrics on $\mathbb{R^n}$

I am working on a problem and want to verify that my logic and reasoning is correct. This is my first time working with metric spaces. Show that the following define equivalent metrics on ...
4
votes
1answer
31 views

Does differentiability imply having bounded variation on some subinterval?

Suppose that $f:(a,b)\to\mathbb{R}$ is a differentiable function. Does it follow that $f$ has bounded variation on some subinterval $[c,d]\subset (a,b)$? Details and ideas Being differentiable ...
0
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1answer
48 views

$f$ convex strictly decreasing function , is $f'(x+\delta)-f'(x)$ convex

Assume you have a strictly decreasing convex differentiable function $f(x)$, $x \in \Bbb R^+$, I am wondering if the increment of the first derivative is also convex; i.e., $$g(x) = f'(x+\delta) - ...
4
votes
2answers
58 views

Show that $\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))dt=\ln(b/a),\,a,b>0$.

Show that $$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))dt=\ln(b/a),\,a,b>0.$$ Thanks to wikipedia I know that $$\int\limits_0^\infty\frac{1}{t}(\cos(at)-\cos(bt))\,dt ...
0
votes
2answers
50 views

If $\{E_\alpha\}$ is connected, $\bigcap\limits_{\alpha\in A}E \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E$ is connected

If $\{E_\alpha\}_{\alpha\in A}$ is connected in $\mathbb{R}^n$, $\bigcap\limits_{\alpha\in A}E_\alpha \neq \emptyset$, then $\bigcup\limits_{\alpha\in A}E_\alpha$ is connected. I have zero intuition ...
5
votes
1answer
43 views

Definition of Sigma Algebra

I was wondering, why are we not allowed to take arbitrary unions (likewise intersections) in the definition of a sigma algebra?; I am looking for a more or less intuitive reason. It seems to me that ...