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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Prove that $d(x,y)=\sum_{i=1}^\infty \dfrac{|x_i - y_i|}{2^i}$ 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
19 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
12 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 ...
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10 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 $k \geq N$ . ...
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16 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 ...
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1answer
17 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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15 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
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0answers
50 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 ...
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1answer
22 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 ...
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6 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 ...
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1answer
15 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
30 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 ...
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1answer
29 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) $. ...
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2answers
22 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 An

I am trying to understand limsup and liminf. I have this homework problem. For each n an element of the Natural Numbers let An=[0,1] if n is odd and An=[1,2] if n is even. Find both ...
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1answer
35 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 ...
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1answer
26 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 ...
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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
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1answer
39 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
33 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 ...
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0answers
8 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
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3answers
30 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
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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
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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 ...
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17 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.
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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. ...
3
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1answer
27 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 ...
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36 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 $ ...
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39 views

Prove this series is convergent. [on hold]

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

Second Fundamental theorem of calculus

I need to use the second Fundamental theorem of calculus to work out: $$\int_{0}^\frac{\pi}{8}\tan(2x)\mathrm dx$$ Firstly it is clear that $\tan(2x)$ is continuous on ...
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1answer
8 views

Separability of the Space of all Real-Valued functions over $[a,b]$ with a Continuous First Derivative

I'm reading Neal Carothers' Real Analysis and I'm stuck on the following question: Let $f$ be real-valued, continuously differentiable function over $[a,b]$ and let $\epsilon>0$. Show that there is ...
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3answers
106 views

$ \sum_{n=1}^{\infty}a_{n} $ diverges but $ \sum_{n=1}^{\infty}\frac{a_{n}}{1+a_{n}^{2}} $ sometimes converges and sometime diverges.

Let $ \lbrace a_{n}\rbrace $ be a sequence of positive terms such that $ \sum \limits_{n=1}^{\infty}a_{n} $ diverges. I am going to show that the series $$ \sum ...
2
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34 views

Boundary on a manifold

I was wondering how I can see if a manifold has a boundary just by looking at the surface? The thing is that I want to understand how to apply the Gauß Bonnet theorem to surfaces and there I need to ...
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0answers
42 views

A book like Michael Spivaks Calculus, for multivariate Calculus.

Is there a book like Michael Spivaks Calculus, that is for Multivariate Calculus? That is a "real analysis" multivariate calculus book?
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32 views

Why is this superior limit independent of $i$ and $j$?

Let $A=(a_{i,j})_{i,j\in E}$ denote a finite irreducible and non-negative Matrix. Then $$ \limsup_{n\to\infty}(a_{i,j}^n)^{1/n} $$ is independent of $i$ and $j$. Here $a_{i,j}^n$ means ...
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43 views

An example of a function for which the equality $M_1 = 2 \sqrt{M_0M_2}$ holds.

Let $f$ be twice differentiable on $(a,\infty),a\in \Bbb R$ and let $$M_k = \sup \{|f^k(x)|\mid x \in (a, \infty) \} < \infty, k=0,1,2.$$ $a)$ Prove that $M_1 \leq 2 \sqrt{M_0M_2}$. $b)$ Give an ...
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15 views

Proof of the Poincare inequality for $W_0^{1,2}((a,b))$.

I have a question about an exercise for which I already have the solution, which I do not unterstand completely. Let $a, b \in \mathbb R$ with $0 < a < b$. Then we have \begin{align*} ...
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1answer
30 views

Supremum of a continuous function on $[a,b]$

Is the supremum of a continuous function $f$ on $[a,b]$ equal to the maximum of $f$ on $[a,b]$? What about a bounded function on $[a,b]$?
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2answers
65 views

How can I prove every Riemann sum of x^2 of [a,b] is the integral?

I find that to prove $\int_a^b x\,dx = (b^2-a^2)/2$ (using the $\epsilon - \delta$ definition of Riemann integrable) is pretty straightforward, and after manipulating some sums, I end up with an ...
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1answer
47 views

The Laplace transform of the Heaviside function

I am studying complex analysis but, because I'm an engineer, I have a lot of doubts. I'm going to present my doubts and it would be nice if someone helps me to see things clearly. Let's start with ...
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18 views

For $k$ and $l$ real constants, prove: (a) If $l \geq k+1$, $\int^\infty_0$ $e^{-x \frac{sin^k(x)}{x^l}}$ = +$\infty$

For $k$ and $l$ real constants, prove: (a) If $l \geq k+1$, $\int^\infty_0$ $e^{-x \frac{\sin^k(x)}{x^l}}$ = +$\infty$ (b) If $l \geq 1$, $\int^\infty_0$ $e^{-x \frac{\cos^k(x)}{x^l}}$ = ...
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1answer
8 views

Linear homotopy

Let $\lambda, \mu:[a,b]\longrightarrow X\subset\mathbb{R}^n$ paths such that the straight line $[\lambda(s),\mu(s)]$ lies in X for all $s\in[a,b]$. Set: $$\begin{array}{lccc} ...
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40 views

Prove the relation $\frac{1}{x}$=$\int^\infty_0$ $e^{-xt}$ dt, for $x>0$. Use it to prove $\int^\infty_0$ $\frac{\sin(x)}{x}$ dx = $\frac{\pi}{2}$

Prove the relation $$\frac{1}{x} = \int^\infty_0 e^{-xt}\, \text{d}t, \text{ for } x>0.$$ Use it to prove $$\int^\infty_0\frac{\sin(x)}{x}\, \text{d}x = \frac{\pi}{2}.$$ "Hint: Use ...
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17 views

Prove $f^l$ has (k-l) zeros

I'm trying to solve this exercize, I think we should use the Rolle's theorem but I do not see how to proceed. Let $$I \ \text{in} \ \mathbb{R} \ , (l,k,n) \in \mathbb{N}^3 \text{such as} \ 0\leq ...
4
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2answers
102 views

Show that $\int^\infty_0$ $\int^\infty_0$ sin($x^2$+$y^2$) dxdy value is $\frac{\pi}{4}$

I am trying to show that the value of $\int^\infty_0$$\int^\infty_0$ sin($x^2$+$y^2$) dxdy is $\frac{\pi}{4}$ using Fresnel integrals. I'm having trouble splitting apart the integrand in order to ...
0
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1answer
37 views

Mean Value Theorem for Integrals

I understand their is an easy way of doing this, I just want to check if my working to a more complicated method is correct. $f$ continuous on $[a,b] \implies $ Riemann Integrable on $[a,b]$ ...
4
votes
1answer
48 views

A rigorous, formal real analysis multi-volume work by an Australian writer

Several years ago I saw at the library a textbook on Real Analysis by some Australian professor that was extraordinarily rigorous, formal and self-contained like nothing I'd seen before or since. It ...
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16 views

Identifying the Dominant terms

I've got the following sequence and I'm trying to work out what the dominant term is. $$a_n=\frac{3n^2+5^n}{2^n-5}$$ now I know that the dominant term is either $5^n$ or $2^n$ as $c^n$ dominates ...
3
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3answers
93 views

Show that $S=\mathbb{Q} \cap [0,2]$ is not compact

Question: Let $S$ be the set of rational numbers in the interval $[0, 2]$. Using the definition of compactness, show that $S$ is not compact. Using the definition of closeness/sequential-compactness, ...