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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11 views

whats the toplogic error in discret mathematics? [on hold]

"Are your siblings older or younger than you?", or perhaps "Are you the oldest?" or "Are you the youngest?" The term birth order would only be used in a scientific context. Even siblings is ...
1
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1answer
7 views

Is the Hölder space with exponent $\beta$ dense in the space with exponent $\alpha$ for $\alpha<\beta$?

For $0<\alpha\leq 1$ let $\Lambda_{\alpha}([0,1])$ be the space of functions on $[0,1]$ such that $||{f_{\Lambda_{\alpha}}}||<\infty$, where ...
2
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0answers
20 views

Inequality About $f(t)=\int_{0}^t \sqrt{\cos(t)} dt$

During my projet, I encountered the following function defined for all $\displaystyle t\in[0,\frac{\pi}{2}]$ by : $$f(t)=\int_{0}^t \sqrt{\cos(t)} dt$$ and I need to prove the inequality below : ...
1
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1answer
6 views

Degrees of Polynomials that Converge Uniformly to a Non-Polynomial

I'm reading Neal Carothers' Real Analysis and there's a problem I'm stuck on: Let $p_n$ be a real-valued polynomial of degree $m_n$, and suppose that $(p_n)$ converges uniformly to a real-valued, ...
1
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3answers
17 views

Compact Sets Metric Spaces

Lets $(\Bbb R,|x-y|)$ be a metric space. By the Heine-Borel theorem, it obviously follows that $\Bbb Q$ is not a compact set. Now, if I were to consider $\Bbb Q \cap[-1,0]\subset\Bbb R$ is that a ...
0
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0answers
11 views

Holder's inequality for improper integral

Does the Holder's inequality hold for improper integral? E.g. $$ \lim_{b\rightarrow \infty} \left|\int_a^b f g d\alpha \right| < \lim_{b\rightarrow \infty} \left( \int_a^b |f|^p d\alpha ...
3
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1answer
38 views

Need help with continuing an idea concerning showing that $4\sum\limits_{n \ge 1} a_n^2 \ge \sum\limits_{n \ge 1} \frac1{n^2}(a_1+…+a_n)^2 $

I recently encountered the following problem: If $\sum a_n^2 $ converges and $\alpha_n= \frac{a_1+...+a_n}{n}$ then show that: $$4\sum_{n \ge 1} a_n^2 \ge \sum_{n \ge 1} \alpha_n^2$$ I had an ...
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0answers
5 views

Differentiability of an extension of the 2-times continuously differentiable function on a half ball

Let $\Omega^+:=\{x=(x_1,\cdots,x_n)\in{\Bbb R}^n\mid |x|<1,x_n>0\}$ and suppose $f\in C^2(\overline{\Omega^+})$ with $f\mid_{\Omega_0}=0$ where $$ \Omega_0:=\partial\Omega^+\cap\{x_n=0\}. ...
-1
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1answer
23 views

prove by using the mean value theorem

If $f$ and $g$ are differentiable on $\mathbb{R}$, $f(0) = g(0)$, and $f'(x) \leq g'(x)$ for all $x \geq 0$, show that $f(x) \geq g(x)$ for all $x \geq 0$.
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19 views

Linear independence of differential 1-forms

Let be $(E,\mathbb{K}),(F,\mathbb{K})$ Banach space and $U\subset E$ open set. If $f_1,...,f_n\in\Omega_1(U;F)$ differential 1-forms are linearly independents, where $$\Omega_1(U;F)=\{f:U\rightarrow ...
0
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0answers
20 views

Determine whether the following expression is positive

I am faced with a problem where I need to show that an expected expression is positive. But I fail to give a strict proof. $$A=E_v ...
0
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1answer
13 views

Properties of sup and lim inf.

Let $(a_{n})_{n \geq1}$ be a sequence of numbers such that $a_n\leq M$ for all $n \geq 1$ . Prove that $$ \lim_{n\to\infty} \inf \{a_n,a_{n+1},...\} = \sup_{n \geq1} \inf\{a_n,a_{n+1},...\} $$ ...
0
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1answer
33 views

Can a real function $f$ on $[0,1]\cap\mathbb{Q}$ be differentiable? [duplicate]

can a real function $f$ on $[0,1]\cap\mathbb{Q}$ be differentiable? if so, and if the derivative of $f$ is zero, then is $f$ is a constant function?
0
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1answer
26 views

Solving the Sequence of this question on Putnam Exam

Problem: Solution: Solution for 2003 A1 Putnam $ka_1 = a_1 + a_1 ... a_1 \le n \le a_1 + (a_1 + 1) + (a_1 + 1) ... (a_1 + 1)$ $= ka_1 + k - 1$ I know these then: What should I do next? Without ...
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3answers
52 views

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals

$\int^\infty_0 \frac{\cos(x)}{\sqrt{x}}\,dx$ Evaluate using Fresnel Integrals (For reference the $\cos$ Fresnel integral is $\int^\infty_0 \cos(x^2)\, dx = \frac{\sqrt{2 \pi}}{4}$) I've tried ...
3
votes
3answers
53 views

What is $\limsup n^ne^{-n^{1.001}}$?

During checking whether or not $\sum_{n=1}^{\infty}{n^ne^{-n^{1.001}}}$ converges, I thought of trying the n-th root test. I got that $\sqrt[n]{n^ne^{-n^{1.001}}}=ne^{-n^{0.001}}$. How can I find ...
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0answers
14 views

Coupling real functions

I ended up with the following two real functions, that are actually the cosine and sine Fourier transform of other more complicated functions: ...
2
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2answers
31 views

Is it possible to find an integer solution $r≥4$ to an equation?

Is it possible to find an integer solution $r≥4$ to this equation? $$11r²³-7r²¹+11r¹⁸-7r¹⁶-2r¹²+11r¹¹- 7r⁹-2r⁷-2 =0$$ I try some special values of $r$ but without any sucess.
5
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2answers
64 views

Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$.

Let $f$ be differentiable on $\Bbb{R}$ and let $\lim_\limits{x\to \infty}f(x)=\lim_\limits{x\to -\infty}f(x)=0$. Prove there exists $a\in \Bbb{R}$ such that $f'(a)=0$. Attempt: If $f$ is constant we ...
3
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2answers
33 views

Prove that $d(x,y)=\sum_{i=1}^\infty \frac{|x_i - y_i|}{2^i}$ converges

Question: Let S be the set of sequences of $0$s and $1$s. For $x = (x_1, x_2, x_3, ...)$ and $y = (y_1, y_2, y_3, ...)$. Define $d(x,y)=\sum_{i=1}^\infty \dfrac{|x_i - y_i|}{2^i}$ Proof the ...
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2answers
25 views

Explanation of the formula $df^{-1} = df\circ f^{-1}.$

Can someone explain the formula (for sufficiently nice $f$), $$df^{-1} = df\circ f^{-1}$$ So far, I have tried working with the relation $df^{-1} = (df)^{-1}$ and the chain rule but I am not able to ...
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1answer
15 views

Function continuity outside a closed subset

Let $f:M \subset \mathbb{R}^p \to \mathbb{R}^q $,continuous at $a \in M $. Show that if $f(a) \notin \overline{B} (b,r) \subset \mathbb{R}^q $, then exists $ \delta > 0 $ such as $ f(x) \notin ...
0
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0answers
12 views

Series, and limits proof. Show $|n b - \Sigma _{k =1}^{n} b_k| \leq \Sigma _{k = 1}^{N} |b_k - b| + M(n - N)$

Can someone please verify the proofs? Anything would help. Let {$b_k$} be a real sequence and $b \in R$. a) Suppose that there are $M,N \in N$ such that $|b - b_k| \leq M$ for all $k \geq N$ . ...
0
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0answers
26 views

about Heine-Borel Theorem in a function space

In Pugh's real mathematical analysis. About the Heine-Borel Theorem in a function space, it states that a subset $\epsilon$ $\in C^0$ is compact if and only if it is closed, bounded, and ...
1
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1answer
21 views

Show that $d_V(x,y)$ is metric

Question: On the set of integers $\mathbb{Z}$, show that the function $d$, defined as follows, is a metric: $$d_V(x,y) = \begin{cases} 0 & \text{if}\ x=y \\ \min{\{\dfrac{1}{n!}}\}\mid\ n!\ ...
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16 views

Xavier spent less than of an hour walking home from school. Which fraction is less 5 2 than ? 5 5 F 7 3 G 4 5 H 10 2 J 9 [on hold]

math is a awesoma and very creative subject.Please help me with this particular question. Thank yo very much
6
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1answer
70 views

if $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$

I have a question $f(x)$ is even and can be infinitely differentiable, how about $f(\sqrt{x})$ in [0,$\infty$)? can we say that the $f(\sqrt{x})$ also can be infinitely differentiable in ...
0
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1answer
25 views

From nowhere dense perfect set to zero measure set.

I know that Cantor set is nowhere dense and perfect. But if I have a nowhere dense perfect set, can I call it a Cantor set? Also, I already proved that a certain subset of the real line is a ...
0
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0answers
8 views

Conditional convergence of series and product

Let $e_k = 0$ for $k$ is odd and $e_k = 1$ when $k$ is even. Set $b_k = \frac{e_k}{k} + \frac{(-1)^k}{\sqrt{k}}$. How do I show that the series $\sum b_k$ diverge while the corresponding product $\Pi ...
0
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1answer
16 views

Freely homotopic but not homotopic

I want to find a example of closed paths freely homotopic but not homotopic (I do not have many tools, like fundamental group, then has to be the simplest way possible). I thought at the following: ...
0
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1answer
34 views

how to proof that this function is zero

given $f$ continuous and diferentiable into $\mathbb{R}$ such that $\forall x\in\mathbb{R},|f'(x)|\le|f(x)|$ and $f(0)=0$ then proof that $f(x)=0$ atempt: taking $x>0$, since $f$ is ...
1
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1answer
30 views

Equality on functions in $ \mathbb{R}^n $

Let $ f,g : M \subset \mathbb{R}^p \to \mathbb{R}^q $ continuous. Given $ a \in M $, supose that all open ball centered in $a$ contains a point $x$ such as $f(x) = g(x) $. Show that $ f(a) = g(a) $. ...
0
votes
2answers
23 views

Continuity of the multiplication map $f\mapsto x^2 f(x)$ between normed spaces

Let $F:C[0,2]\to C[0,2]$ be the map defined by $(F(f))(x)=x^2f(x)$. Show that $F$ is continuous as a function from $(C[0,2],\|\cdot\|_{\sup})$ to $(C[0,2],\|\cdot\|_{2})$. I read this solution: ...
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0answers
18 views

Finding limsup and liminf for odd and even $A_n$

I am trying to understand $\limsup$ and $\liminf$. I have this homework problem: For each natural number $n$, let $A_n=[0,1]$ if $n$ is odd, and $A_n=[1,2]$ if $n$ is even. Find both ...
1
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1answer
36 views

Real analysis, sequences and inequalities

Let $(x_n)$ be a sequence such that $$ \lim x_n = \alpha $$ What would this limit be: $$ \lim \frac{ x_1 + ... + x_n }{n} = $$ ?? I feel like the limit is again $\alpha $, but I don't know how ...
1
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1answer
27 views

Example of closed unit ball?

I am not understanding the concept of ball on a set $E$ and closed unit ball $B_1$ in $B(E)$. I need to prove or disprove by example that if the closed unit ball $B_1$ is compact or not in a metric ...
4
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27 views

Real analysis question about proof of convergence

Let $(x_n)$ be a sequence such that $x_n \to 0 $ and let $(y_n)$ be a sequence such that $(y_n)$ is bounded. Please show that $(x_ny_n) \to 0 $ try Since $(y_n)$ is bounded, we can find some $\alpha ...
3
votes
1answer
41 views

Finding distance between the unit ball in $\mathbb{R}^2$ and the point $(1,1)$

Given the euclidian metric. $d(x,y) = ((x_1-y_1)^2 + (x_2 -y_2)^2)^{1/2}$ find the distance between the point $(1,1)$ and the set $A = \{x=(x_1,x_2) \in \mathbb{R}^2 : x_1^2 +x_2^2 \leq 1 \}$ Where ...
2
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2answers
35 views

Why upperbound $|x-a|$ by 1 in the proof of continuity?

In most (all?) proofs of continuity of polynomials ($x^2, x^3$, etc), for example in Max Rosenlicht's book (http://www.math.pitt.edu/~frank/pittanal2121.pdf, page 97), the usual trick is to get to the ...
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2answers
28 views

If $p(x) = x^3 + ax^2 + bx + c$ then $p:\mathbb{R} \rightarrow \mathbb{R}$ is a homeomorphism if and only if $a^2 \leq 3b$

I know $p$ is surjective and continuous, but I'm not sure how that inequality $a^2 \leq 3b$ is gonna help me with the function being injective and its inverse being continuous. Can anyone give me a ...
0
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0answers
10 views

How would you find out the cartesian equation tangent to two surfaces at given point?

I am given two surfaces and asked to find out a pair cartesian equations that are tangent to these two surfaces. I know how to find out tangent plane to one surface. But, how would you find out a ...
5
votes
2answers
31 views

Does the limit of this double sequence exist?

Consider $$a_{mn}=\frac{m^2n^2}{m^2+n^2}\left(1-\cos\left(\frac{1}{m}\right)\cos\left(\frac{1}{n}\right)\right)$$ Does $\lim_{m,n\to\infty}a_{mn}$ exist? It can be seen that ...
2
votes
1answer
21 views

Preservation of inequality on continuous functions

Let $ f,g:M \subset \mathbb{R}^{p} \to \mathbb{R} $ countinuous function at $a \in M$. Show that if $f(a) < g(a)$ then exists $ \delta >0 $ such as for $x$ and $y$ in $M \cap B(a, \delta) $ ...
2
votes
2answers
49 views

Proving that a polynomial has a positive root

So I want to prove that a polynomial $ P(x)=a_nx^n+a_{n−1}x^{n−1}+.....+a_1x+a_0 $ has a positive root. I'm given that $ a_n $ is positive and $ a_0 $ is negative. I want to know how to apply the ...
0
votes
0answers
18 views

Product of measure spaces

Show that B(R^n)=B(R)*B(R)*B(R)...n times where B(R) is a Borel sigma algebra of R. I know B(R^n) subset of B(R)*B(R)*B(R).. But I couldn't get idea of reverse inclusion. Please help me out.
0
votes
0answers
26 views

Calculate the diameter of the unit ball in $\mathbb{R}^3$ using the Euclidean metric.

So the question states, Let $B = \{x = (x_1,x_2,x_3) \in \mathbb{R}^3: x_1^2 +x_2^2 +x_3^2 \leq 1 \}$ be the unit ball in $\mathbb{R}^3$. Compute the diameter of $B$ for each of the following metrics. ...
4
votes
1answer
29 views

Theorem 3.1 from Milnor's Morse Theory

Milnor is in the business of proving that if $f: M \to \mathbb{R}$ is a smooth function, $a < b$, and $f^{-1} ([a,b])$ is a compact subset of $M$ containing no critical points, then $M^a$ is ...
1
vote
0answers
45 views

Existence of a constant $C$ and a sequence $ x_1, x_2, …\in [0,1] $ [on hold]

a) Prove that for every sequence $x_1, x_2, ...\in [0,1]$ there exists some $C>0$ such that for every positive integer $r$ there are positive integers $m,n$ satisfying $ |n-m|\geq r $ and $ ...
-1
votes
2answers
41 views

Prove this series is convergent. [on hold]

Prove this series is convergent. $$0-\frac{1}{2}+\frac{1}{2^{2}}-\frac{1}{3}+\frac{2}{3^{2}}-\frac{1}{4}+\frac{3}{4^{2}}- ...$$
0
votes
2answers
114 views

Second Fundamental theorem of calculus

Theorem 3.20 (Second Foundamental Theorem of Calculus) Let $f$ be a continuous function on $[a, b]$ and $F$ any function on $[a, b]$, differentiable on $(a, b)$, continuous on $[a, b]$ such that ...