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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80 views

Continuous injective map is strictly monotonic

Show that if $f: \mathbb{R} \to \mathbb{R}$ is a continuous injective map, then it is strictly monotonic. Could someone give me a proof for this? I have the intuition for why it's true - I'm just ...
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2answers
52 views

if one is bounded the other one is also bounded…

If $(a_n)$ is a bounded sequence of positive rational numbers and equivalent with sequence $ (b_n)$, show that $ (b_n)$ is also bounded. I was thinking that since $(a_n)$ and $ (b_n)$ are equivalent ...
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3answers
394 views

You use Rolle's theorem to prove n-polynomial has n distinct roots, but how would you do this question?

Show that if a polynomial of degree n has n real roots (which don't have to be distinct) then its derivative has n-1 real roots (which also don't have to be distinct).
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1answer
118 views

LogSine Integrals $\int_0^{\pi/3}\theta \ln^2\big(2\sin\frac{\theta}{2}\big)d\theta$.

Hi this will soon end my posts on Log Sine integrals, and we can progress into other classes of integrals. The log sine integral I am trying to calculate is given by $$ ...
2
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2answers
92 views

A basic question on measurability of lim sup and lim inf of a function

Suppose $f: \Bbb R \to \Bbb R$ is a Borel measurable function. I have to prove that $\{x: $f$ \text{ is discontinuous at } $x$\} \in B(\Bbb R)$. So, I am trying to prove that the complement event i.e. ...
2
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0answers
128 views

Is this sequence a Cauchy sequence?

Consider a continuous mapping $f: X \rightarrow X$, where $X \subset \mathbb{R}^n$ is compact and convex. Consider a sequence $\{x_k \in X\}_{k \geq 0}$ such that for all $k \geq 0$ and $h \geq 1$, ...
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1answer
109 views

$\frac{5\pi^3}{154}=\int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta$

I am trying to prove $$ \int_{\pi/6}^{\pi/2}\bigg[\Re\big(\text{Li}_2(4\sin^2\theta)\big) +\text{Li}_2\bigg(\frac{1}{4\sin^2\theta}\bigg) \bigg]d\theta=\frac{5\pi^3}{54}. $$ Clearly, this closed form ...
2
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3answers
344 views

How to parametrize a curve by its arc length

I am reading on Wikipedia that ''...Any regular curve may be parametrized by the arc length (the natural parametrization) and...'' I know that if $a(t) = (x(t),y(t),z(t))$ is a curve (say, smooth) ...
2
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2answers
72 views

How to prove convergence of $a_n$ if $(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$?

Could you give me some hint how to conclude convergence of $a_n$ from this feature : $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ From $$(n+1)(a_{n+1}-a_n)=n(a_{n-1}-a_n)$$ we may conclude that ...
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0answers
43 views

Curve with second derivative identically zero

I just solved the following exercise: Let $I = (a,b)$ be an open interval, $a,b$ possibly equal to $\pm \infty$ and $\alpha: I \to \mathbb R^3$ a smooth parametrized curve that has the property that ...
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0answers
34 views

Why is it an interior minimum

I just solved the following exercise: (Here $I=(a,b)$ is an open interval, possibly $a,b=\pm \infty$ and $\alpha : I \to \mathbb R^3$ is smooth.) Let $\alpha$ be a parametrized curve that does not ...
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1answer
68 views

Questions on Continuous Function

I know that it is very obvious that intuitively, a continuous function cannot have any gap in between. However, I am having difficulty proving it. Normally, in textbook and also in my real analysis, ...
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1answer
52 views

functions of two variables with one variable defined on a compact set uniformly converge to zero

Let $f$ be a holomorphic function on $[0,1]\times \mathbb{R}$. If for each $x\in [0,1]$ fixed, $\lim_{y\to\infty}f(x,y)=0$, prove that $f$ is bounded. My idea: I do not know how to prove and I also ...
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3answers
113 views

Bolzano - Weierstrass theorem for sequences

There is couple of things that i don't understand in the proof that Every bounded sequence has a convergent subsequence here is the textbook proof (first part) proof: let $(s_n)$ be a sequence ...
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3answers
106 views

Taylor expansion for a multivariable function

\begin{align} T(x_1,\dots,x_d) &= \sum_{n_1=0}^\infty \sum_{n_2=0}^\infty \cdots \sum_{n_d = 0}^\infty \frac{(x_1-a_1)^{n_1}\cdots (x_d-a_d)^{n_d}}{n_1!\cdots n_d!}\,\left(\frac{\partial^{n_1 + ...
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2answers
60 views

Multiplying and Dividing Series

For example, how do you compute the taylor series for $$e^x \sin x=\sum_{n=0}^{\infty} \frac {x^n}{n!} \sum_{n=0}^{\infty} (-1)^n\frac {x^{2n+1}}{(2n+1)!}$$ Of course I want the result to contain ...
1
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1answer
33 views

Question regarding standard proof to: If f is continuous on [a,b] then f is bounded on [a,b].

The standard proof first builds up a sequence by saying that for each integer n there exist a number $x_n$ in [a,b] such that |$f(x_n)$| > $n$. Then, it proceeds to use the Bolzano weierstrass theorem ...
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2answers
136 views

how to show integral is continuous

I am working on a problem: Let $f: [a,b]\to R, a<b$, be Riemann integrable on $[a,b]$. Show that $$\int_a^bf(x)dx=\lim_{T\to b}\int_a^Tf(x)dx.$$ Letting $F(x)=\int_a^x f(t)\,dt$, I think this ...
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3answers
82 views

Developing an intuition for compact and open sets

I'm having trouble picturing what compact sets and open sets actually are. Open and closed intervals make enough sense to me, but for whatever reason, moving to the next level of abstraction is ...
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0answers
67 views

LogSine Moments $\int_0^\sigma \theta^k \ln^{n-1-k}\big| 2\sin\frac{\theta}{2}\big|d\theta$

This integral is known as the moments for the generalized log-sine integrals. The notation I am using is similar to Lewin and what he used in the 1950's-1980's. $$ ...
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1answer
84 views

LogSine Generating Fn $ \int_0^\pi \big(2\sin\frac{\theta}{2}\big)^x e^{\theta y} d\theta$

This is related to generating functions for Ls (Log Sine Integrals.) I am trying to calculate $$ \int_{0}^{\pi}\left[2\sin\left(\theta \over 2\right)\right]^{x} {\rm e}^{\theta y}\,{\rm d}\theta. $$ ...
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1answer
200 views

LogSine Integral $I=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) d\theta$

These are known as LogSine integrals at $2\pi/3$, so I will call the integral Ls as this is common in the literature. I am trying to prove $$ Ls=-\int_0^{\pi/3} \ln^2\big(2\cos \frac{\theta}{2}\big) ...
3
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0answers
100 views

Density of a set of functions in Schwartz space

I have a difficulty doing the following problem: Let $S(\mathbb{R}^n)$ be the Schwartz space. I need to determine whether the following set of functions $A$: $$A= \{f\in S(\mathbb{R}^n): ...
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0answers
48 views

Extracting coefficients from a transformed generating function

Let $G(z)=\sum_{k\geq 0} a_kz^k$ be a generating function such that $z^aG(1-z)=P(z)$, where $P(z)$ is a polynomial and $a$ is a positive integer. I'm interested in $P(z)[z^n]$, the coefficient in ...
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0answers
39 views

Can we just use Rolle's theorem to prove Taylor's Theorem, or is there another way?

The following real function, $h(x)$, is twice differentiable and $a$, $b$ are distinct real numbers: $$h(x)=f(b)-f(x)-(b-x)f'(x)+\frac{K(b-x)^2}{2}$$ where the constant $K$ is chosen so that ...
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1answer
467 views

Help with epsilon-delta proof that 1/(x^2) is continuous at a point.

I'm trying to prove that $\lim_{x \to x_0} \frac{1}{ x^2 } = \frac{1}{ {x_0}^2 }$. I know this means that for all $\epsilon > 0$, I must show that there exists a $\delta > 0$ such that $\left | ...
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0answers
39 views

Theorem proof of local inclusions

Please help prove the following theorem: Let $f$ be a map from $R^l$ to $R^m$ that is continuously differentiable on an open set that contains a point $p.$ Suppose that the linear map $J_f(p)$ ...
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2answers
56 views

What does it mean for two numbers to be $\epsilon-$close?

I found on my textbook [...] so those two numbers are $\epsilon-$close. QED. What does it mean for two numbers to be $\epsilon-$close?
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0answers
69 views

The implicit function theorem

Prove that if the functions $f_1,...f_n$ in the statement of the implicit function theorem are assumed to be $k$ times differentiable (i.e., all partial derivatives of order k exist and are ...
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5answers
222 views

Mean Value Theorem: Continuous and Differential

Mean Value Theorem. If $f:[a,b] \rightarrow R$ is continuous on [a,b] and differentiable on (a,b), then there exists a point $c \in (a,b)$ where $$f'(c) = \frac{f(b)-f(a)}{b-a} $$ I am having a ...
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1answer
44 views

How to define a continuous function $f:I\times I\longrightarrow I$ such that $f(0, t)=t$ and $f(1, t)=1$ for all $t\in I$?

Let $I=[0, 1]$. I need some help to define a continuous function $f:I\times I\longrightarrow I$ such that $$f(0, t)=t\quad \textrm{and}\quad f(1, t)=1$$ for all $t\in I$. The nearest I got was: $$f(s, ...
0
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1answer
84 views

Proof about complex exponential function forming an infinite dimensional vector space

Okay, so I sort of understand what is going on here. this has to do with the fact that the exponential function is bijective over this interval, yes? Either way, I have no idea where to start with ...
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0answers
65 views

Mean value theorem mindset

So I am to learn to use the mean value theorem to prove these types of problems that I will list. I would really like for someone to provide some visual/intuitive information on how I can imagine the ...
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1answer
55 views

Showing that some differential equation has an infinite dimensional solution space?

I don't see how to proceed or even where to start to show this thing that I have found: The differential equation $$(\sin x)\frac{dy}{dx} - 2(\cos x)y = 0$$ has an infinite solution space of ...
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2answers
33 views

Show this function can be defined as the limit function

Let f: $ \mathbb{R} \rightarrow \mathbb{R} $ be defined by f(x) = 1 for x $\in \mathbb{Q} $, f(x) = 0 otherwise. We can see f is not regulated. Show that f may be obtained as a limit function: f(x) = ...
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2answers
75 views

First Order Lagrange Remainder Using Ordinary MVT

I would like to show my Calc I class that $f(x)=f(a)+f'(a)(x-a)+(f''(c)/2)(x-a)^2$ for some $c$ in $(a,x)$ (for $f$ smooth). Just from the form of the statement, it seems as though this should be ...
0
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1answer
35 views

Uniform and Pointwise convergence of $\{f_n(x)\}$ where $f_n(x) = x^n \cos(2\pi xn)$?

Consider the sequence of functions $\{f_n(x)\}$ where $f_n(x) = x^n \cos(2\pi xn)$ and $x \in [0,1]$ What is the point-wise limit $g(x)$ of the sequence? Does the sequence converge uniformly to ...
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1answer
75 views

$E$ measurable if and only if $E \cap (a,b)$ is measurable for any interval $(a,b)$

We take the definition of measurability to be the following: $E \subseteq \mathbb{R}$ is measurable if for any $\varepsilon > 0$ there is an open set $G$ and a closed set $F$ such that $F \subseteq ...
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1answer
62 views

Real Analysis question dealing with Intermediate Value Theorem. Or is there another way to do it?

The function $g: (0, \infty) \to \mathbb{R}$, is continuous, $g(1)>0$ and $$\lim_{x \to \infty} g(x) = 0$$ Show that for every $y$ between $0$ and $g(1)$ the function takes on a value in $(\ y,\ ...
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2answers
189 views

Prove $f $ is identically zero

$f:\Bbb R \to\Bbb R $ is differentiable, $f(0)=0$ and $|f'(x)|\le|f(x)|$ for all $x$ then prove $f$ is identically zero. I tried to use mean value theorem and end up in $|f(x)|\le |x||f(c)|$ for some ...
0
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1answer
28 views

$\sum_{n=1}^{\infty}|a_{n} x^{n}| < \infty \implies \sum_{n=1}^{\infty} |a_{n}| (|x|^{n-1}+ |x^{n-2}y|+…+|y^{n-1}|) <\infty $?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$. Now, we let, $x, y \in \mathbb R$ with $x\neq ...
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3answers
49 views

$\sum_{n=1}^{\infty} a_{n}= \sum_{n=1}^{\infty}b_{n}$ and $\sum_{n=1}^{\infty} |a_{n}| < \infty \implies \sum_{n=1}^{\infty} |b_{n}| < \infty$?

Suppose that $\{a_{n}\}_{n\in \mathbb N}, \{b_{n}\}_{n\in \mathbb N} \subset \mathbb C$ such that both the series, $\sum_{n=1}^{\infty} a_{n}$ and $\sum_{n=1}^{\infty} b_{n}$ converges, and its sum ...
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1answer
49 views

How to justify, $\sum_{n=1}^{\infty} a_{n} x^{n} - \sum_{n=1}^{\infty}a_{n}y^{n}=\sum_{n=1}^{\infty} a_{n} (x^{n}-y^{n})$?

Let $\{a_{n}\}_{n\in \mathbb N} \subset \mathbb C$ so that the series, $\sum_{n=1}^{\infty} a_{n} x^{n},$ converges absolutely for all $x\in \mathbb R$ and we let $K_{1}$ be a compact subset of ...
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1answer
63 views

What is the formal meaning of “determine” in Baby Rudin 2.40?

In Theorem 2.40, Rudin talks about a $k$-cell $I$ formed by the intervals $[a_1, b_1], \ldots, [a_k, b_k]$. We split each interval at its midpoint $c_j = \frac{a_j + b_j}{2}$ and end up with $2k$ ...
0
votes
1answer
73 views

Question on Martingales and Brownian Motion

I've run into an issue and I am unsure of how to proceed. In the text I'm working through, the following is left "as an exercise to the reader." Normally proofs listed as such tend to be fairly simple ...
0
votes
2answers
89 views

A basic question on Riemann sum

Suppose $f$ is a non-negative Riemann integrable function in $[a,b]$. Is this true that $$ \sup_P \sum_{j=1}^{n} |f(c_j)(x_j-x_{j-1})| = \int_{a}^{b} |f(x)|dx$$ where $c_j \in [x_{j-1}, x_j]$. I ...
1
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0answers
73 views

The product of two rational Dedekind cuts

If $a,b\in \mathbb{Q}$ and $C_a$ and $C_b$ are both positive rational Dedekind cuts then $C_a\cdot C_b=C_{a\cdot b}$. First of all this is my definition of product: Let $r,s$ Dedekind cuts such ...
2
votes
3answers
43 views

Find the limit of a function??

Determine the following limit: $$\lim\limits_{x\to0}\frac{(1+x\cdot2^x)^{\frac{1}{x^2}}}{(1+x\cdot3^x)^{\frac{1}{x^2}}}$$
7
votes
1answer
63 views

The Limit: $\lim_{x \to \infty}\frac{e^{f(x+a)}}{e^{f(x)}}$

I'm doing some challenge review problems and I was wondering whether this proof looked correct: Suppose that $f: \mathbb{R} \rightarrow \mathbb{R}$ be a differentiable function with $\lim_{x \to ...
1
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1answer
75 views

Gaining an intuitive understanding of measure & sigma-algebras

Taking my first course in measure theory. Consider an example where $\Omega$={all integers from 1 to 16}={1,...,16} where classes of sets are defined by $C_1$={1, 2, 3, 4, 5, 6, 7, 8} $C_2$={9, 10, ...