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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Uniform Continuity of Cos Function without use Lipschitz Condition

To show that the cos function is uniform continuous on R, instead of showing that the function satisfies that Lipschitz condition ($|cos(x_1)-cos(x_2)| \le |x_1-x_2|$ for all $x_1, x_2$ in $R$), is ...
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99 views

Convergence of $\int \cos(nx)$

How can we show $\lim_{n\rightarrow\infty}\int_X\cos(nx) \rightarrow 0\;$? (X can be any set)
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1answer
64 views

Showing Riemann integrability

Let $f: [a,b] \to \mathbb{R}$ be bounded. Show that $f$ is Riemann intergrable iff $$\bar{\int_{a}^{b}} f = -\left[\bar{\int_{a}^{b}} -f\right]$$ My attempt is as follows. "$\Leftarrow$" ...
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196 views

Transcendence of $e$ (proof)

I'm trying to get through the proof of transcendence of $e$ (the base of the natural logarithm) already for a couple of days, but now I got seriously stuck. Proof is in most sources roughly the same. ...
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2answers
226 views

Trying to show that a function is zero almost everywhere given a constraint on its Lebesgue measure.

We have that $g$ is a measurable and bounded function on $[a,b]$. I have $\int_a^cg=0$ for every $c\in[a,b]$. I want to show $g=0$ on $[a,b]$ except possibly on a subset of measure zero. Proof. By ...
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Cantor set - a question about being metrizable and about the connected components

I have a question regarding Cantor set given to me as a homework question (well, part of it): a. Prove that the only connected components of Cantor set are the singletons $\{x\}$ where $x\in C$ ...
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4answers
125 views

Showing that $\frac{1+nx^2}{(1+x^2)^n}$ converges to $0$

I aim to show that $\{a_n\} = \frac{1+nx^2}{(1+x^2)^n}$ converges to $0$. The following two facts seem obvious: (1) $\forall n \in \mathbb{N}, a_n \ge 0$ (since each $a_n$ is the product of two ...
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268 views

supremum, infimum - proving

Having given set $$A=\left(\frac{m^2-n}{m^2+n^2}: n,m \in \mathbb{N}, m>n\right) $$ the task is to find its supremum and infimum. So I take $$f(m,n)=\frac{m^2-n}{m^2+n^2}\Rightarrow f(m,1) = ...
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21 views

$g\in \mathcal{L}^{loc}(\mathbb{R}),\ \phi\in\mathcal{C}^\infty_c (\mathbb{R})\stackrel{?}{\Rightarrow} g\phi \in \mathfrak{R}(\mathbb{R})$

Let $g:\mathbb{R}\longrightarrow\mathbb{R}$ be a locally integrable function and $\phi\in\mathcal{C}^\infty_c(\mathbb{R})$ (i.e. $\phi $is a $\mathcal{C}^\infty$ function with compact support in ...
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Proof help needed (related to Bolzano-Weierstrass theorem)

I have an exam tomorrow and could not solve the question below. Could you please help me? Regards Let $(x_n)$ be a bounded sequence, and let $c^∗$ be the greatest cluster point of $(x_n)$. (a) ...
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153 views

parametrizing quarter of a circle

I am given the circle whose equation is: $(x-\frac{1}{2})^{2}+(y+\frac{1}{2})^{2}=\frac{1}{2}$. So, the coordinates of the origin of the circle are: $(\frac{1}{2},-\frac{1}{2})$ and the radius of the ...
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1answer
57 views

Reference request on analysis in $\mathbb{R}$

I need a reference on theorems unique to $\mathbb{R}$, things that go away in higher dimensions. For example: in Topology and Groupoids, it is said that a continuous injective function $f: (a, b) \to ...
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66 views

Disjoint balls on Euclidean space

I'm doing some self-study and I'm stuck on a proof. Prove that if two open balls on $N$-dimensional Euclidean space are disjoint then $d(x,x') \ge r + r'$ $x$ and $x'$ are the centers of the balls, ...
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93 views

Help with proof of a converging sequence.

I am having a hard time understanding this proof (leading up to Newtons method) about why a sequence must converge. English is not my native language so I will do my best to use the correct ...
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770 views

Radon-Nikodym derivative of product measure

For $j=1,2$, let $\nu_{j},\mu_{j}$ be $\sigma$-finite measures on $(X_{j},\mathcal{M}_{j})$ such that $\nu_{j}\ll\mu_{j}$. I want to show that $\nu_{1}\times\nu_{2}\ll\mu_{1}\times\mu_{2}$ and that ...
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550 views

Supremum and infimum: proving with definition

I tried to solve following task, but I am not sure whether my solution is correct. Quest: Find supremum and infimum $$A=\left\{\frac{n - k^2}{n^2 + k^3}:n,k \in \mathbb{N} \right\}$$My attempt: By ...
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1answer
101 views

An example of a series in which using $\limsup$ rather than $\lim$ in performing the root test or the ratio test would make a difference?

I've learned to apply these tests before in Calculus, but in the textbook that I used, the numbers of interest for the root and ratio tests were presented as $\lim_{n\rightarrow \infty}|a_n|^{1/n}$ ...
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1answer
102 views

Convergence of a matrix

Let $A_j$ be a sequence in $\mathbb {C}^{n\times n}$. Show that $\displaystyle \sum_{j=0}^\infty A_j$ converges if $\displaystyle \sum_{j=0}^\infty ||A_j||$ does. Note that $\displaystyle \Vert ...
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1answer
186 views

$C$-doubling $\Bbb R^2$ measure gives measure zero to a straight line?

A metric space $X$ with metric $d$ is said to be doubling on $\Bbb R^2$ if there is some constant $C > 0$ such that for any $x \in X$ and $r > 0$, the Euclidean ball $B(x, r) = \{y:|x − y| < ...
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1answer
303 views

Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable?

Let $E$ be measurable and define $f:E\rightarrow\mathbb{R}$ such that $\{x\in E : f(x)>c\}$ is measurable for all $c\in\mathbb{Q}$, is $f$ measurable? There are a number of equivalent definitions ...
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Open sets and closed sets proof

Theorem A set E is open if and only if the complement $E^{c}$ is closed Proof $E$ open $\iff$ any point $x \in E$ is an interior point $\iff \forall x \in E, \exists$ a ...
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Prove that function f is bounded below

$f:\mathbb{R}^k \rightarrow \mathbb{R}$ is differentiable function such that $\sum_{i=1}^{k} x_i \frac{df}{dx_i}(\mathrm{x} )\ge 0$ for $\mathrm{x}=(x_1,x_2,...,x_k) \in \mathbb{R} ^k$. Prove that ...
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240 views

Why do we need the axiom of completeness? Why won't Cantor's diagonalization work without it?

Okay so for my upcoming test I need to "be able to explain at least one result that would not hold if the axiom of completeness were not accepted" My teacher suggested that I could try to explain why ...
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Question regarding Nested Interval Theorem

Theorem Consider a family of closed intervals, $I_1 = [a_1, b_1], I_2 = [a_2, b_2], \ldots$. If $a_n \leq a_{n+1}$ and $b_{n+1} \leq b_n$ for all $n$ then there is an $x$ which is in every ...
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Limits of solutions to $x\sin(x)=1$ [duplicate]

Possible Duplicate: Sequence of solutions to $x\sin x=1$ Let $0<a_1<a_2<a_3< \dots$ be the positive solutions to the equation $x \sin (x) =1$. What is the value of: $S = \lim ...
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32 views

Limit of Function that Satisfies Simple Property [duplicate]

Possible Duplicate: Limit of a continuous function Let $f$ be a function that is continuous and real on $[0, \infty]$ such that $\lim_{n \to \infty} f(na) =0$ for all $a>0$. What can be ...
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Continuity On $\mathbb{Q}$

I need a continuous function $f:\mathbb{Q}\rightarrow \mathbb{R}$ and discontinuous $g:\mathbb{R}\rightarrow \mathbb{R}$ s.t $f(x)=g(x)$ for all rational $x$ s. So if I say $f(x)=0$ and $g(x)=0$ for ...
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Differentiable $\mathbf{f}:\Bbb R \to \Bbb R^3$, $|\mathbf{f}(t)| = 1$ implies $\mathbf{f}(t)\cdot\mathbf{f}'(t) = 0$

This question comes from Rudin's PMA problem 9.13. For a differentiable function $\mathbf{f} : \Bbb R \to \Bbb R^3$ and $|\mathbf{f}(t)|=1$ for all $t$, I want to prove that ...
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Convergent sequence sufficiency

Definition: a real sequence is a mapping $f : N^+ \mapsto R$ The real sequence $\frac{1}{2^n}$ converges to $0$. If $\epsilon$ in $R^+$ is given, then $\left| \frac{1}{2^n} \right| < \epsilon$ ...
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Question about definition of Oscillation

The oscillation of $\omega_f(A)$ of $f$ on a set $A$ to be the number $$\omega_f(A)=\sup\limits_{x,y\in A}|f(x)-f(y)|=M_A(f)-m_A(f).$$ The following equality is where I'm scratching my head a bit: ...
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covering lemma for finite collection

Covering Lemma: There exist $c\in \mathbb{N}$ ($c=2$ works) s.t. for any finite collection of open intervals $\{I_n\}$ there is a sub collection $\{I_{n_j}\}$, s.t. $\{I_{n_j}\}$ covers same set that ...
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165 views

Can there be a embedding from the real number line to the plane such it is both closed and bounded?

Just a little clarification on the question: when I say line I am essentially referring to the real number line and the plane being $\mathbb R^2$. I am really new to the notion of embedding; hence, I ...
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1answer
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Does Newton's method for inverting a series work?

Suppose we have $z=f(x)$ with $f$ an infinite series. We want to find $f^{-1}(z)=x$. Newton proposed the following method (as described in Dunham): First, we say $x=z+r$. We find $z=f(z+r)$, drop all ...
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134 views

Landau symbols and little o

I was wonderig if the following is true: $o(x^n+x^m)=o(x^n)+o(x^m)$ for $x\to 0$. I tried this way: suppose $m>n$ and let first $f=o(x^n+x^m)$. Then $$\frac{f}{x^n+x^m}=\frac{f}{x^n+o(x^n)} ...
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Coinciding open sets

I'm given two distances that are defined on some metric space and I need to show that open sets and Cauchy sequences coincide for the two distances. What does this mean? I'm avoiding giving details on ...
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1answer
102 views

Neighborhoods are open proof

Let $r' = r - a$ I am trying to proof that neighbourhoods or open balls is an open set. I've watched this video on youtube where the guy explains how the triangle inequality comes into play. I ...
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0answers
86 views

Integration on $R^n$

Let be $K(x,y)=\frac{2x_n}{n\alpha(n)}\frac{1}{\vert x-y\vert^n}$ where n is the dimension and $\alpha(n)$ is the volume o unitary sphere in $R^n$, $x=(x_1,...,x_n)\in R^n$ and ...
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Show the usual Schwartz semi-norm is a norm on the Schwartz space

Let $f \in C^\infty(\mathbb R)$. Define the semi-norm $$ \|f\|_{a,b}=\sup_{x \in \mathbb R} |x^af^{(b)}(x)| $$ where $a,b \in \mathbb Z_+$, and $f^{(b)}$ is the $b$-th derivative of $f$. Show ...
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Conditions for continuous extension of a function on an open set to its closure

Suppose $U$ is a open set in $\Bbb R^n$, and suppose $f\colon U\to \Bbb R$ is a continuous function. Suppose that $f$ is uniformly continuous on every bounded subset of $U$. Question: Can $f$ be ...
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2answers
250 views

Existence of maximum value of a discontinuous function

If $f(x)$ takes a finite real value for all $x$ on the closed interval $[a,b]$, must there be a real number $M$ such that $M\geq f(x)$ for all $x$ on this interval? It seems that if not, there must be ...
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1answer
130 views

A quantitative version of the Weierstrass' Approximation Theorem

Assume that $f\in\mathcal{C}^0([0,1])$. By using Chebyshev Polynomials, it is possible to show that there exists a sequence of polynomials $\{p_n(x)\}_{n\in\mathbb{N}}$ such that: $$ ...
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1answer
252 views

Compute $\sum_{i,j=1}^{\infty} \frac{(-1)^{i+j}}{i^2+j^2}$

Compute $$\sum_{i,j=1}^{\infty} \frac{(-1)^{i+j}}{i^2+j^2}$$
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2answers
167 views

How to use the dense theorem to prove this exercise?

Let $\frac{2n}{n+1}\leqslant p<n,\quad q=\frac{np}{2n-p},\quad u\in L^1(R^n)\cap W^{1,p}(R^n)$, then prove $u^2\in W^{1,q}(R^n).$ I hope someone can show me how to prove it by dense theorem ...
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53 views

Explanation about Goldstein steps

Goldstein in "Classical Mechanics" (1ed) obtains $$-\int \sum_j \left(\frac {\partial V}{\partial q_j} \delta q_j + \frac{\partial V}{\partial \dot q_j} \delta \dot q_j\right) dt=-\delta \int V dt$$ ...
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2answers
97 views

Evaluate integral $\int\limits_1^\infty x^{y^m}dy$

What would be the evaluation of the definite integral $$\int\limits_1^\infty x^{y^m}dy$$ where $x$ with $0 < x < 1 $ is real ; $m > 1$ is any integer. At least an approximation will suffice ...
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graph is dense in $\mathbb{R}^2$

I was asked in a exam: does there exist a function(need not be continous) $f:\mathbb{R}\rightarrow \mathbb{R}$ whose graph is dense in $\mathbb{R}^2$? I proved that graph of a discontinuous linear ...
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1answer
54 views

At most countable subsets of a compact metric space.

As written, the question is: Let (X,d) be a compact metric space. Prove that for each $\epsilon>0$ there exists a positive integer $N$ such that for each $S \subseteq X$, if $S\thicksim Z_N$, then ...
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2answers
57 views

Cases when the Intermediate Value Theorem is true - Part 2

I previously asked this question and was told that an answer is certainly possible but I am still looking for an example. The question was for cases when the intermediate value theorem is true and a ...
4
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1answer
218 views

Function which preserves closed-boundedness and path-connectedness continuous?

Let $f:\mathbb{R^n}\to\mathbb{R^m}$ be a function such that the image of any closed bounded set is closed and bounded, and the image of any path-connected set is path-connected. Must $f$ be ...