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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0
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19 views

Does this theorem concerning upper and lower bnound of a monotone decreasing function have a formal name?

This is the theorem: Let $g$ be a monotone decreasing function and let $a,b \in \mathbb{N}$. Then the following holds true: $$\int_{a}^{b+1}g(x)dx \overset{(i)}{\leq} ...
10
votes
3answers
165 views

Calculating $\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$

Do you see any fast way of calculating this one? $$\int_0^{\infty } \left(\text{Li}_2\left(-\frac{1}{x^2}\right)\right)^2 \, dx$$ Numerically, it's about $$\approx ...
4
votes
0answers
60 views

A discontinuous function $f: X \rightarrow Y$ satisfying: for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$

Find a function $f: X \rightarrow Y$ between metric spaces $X$ and $Y$ that is not continuous but has the property that for each closed ball $B$ of $Y, f^{-1}(B)$ is closed in $X$ Solution Attempt: ...
0
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3answers
44 views

Question about limit of an integral of a continuous function

Let $ f$ be a real-valued continuous function defined for all $0\leqslant x\leqslant 1$, such that $f(0) = 1$, $f(\frac{1}{2}) = 2$ and $f(1) = 3$. Show that $$\lim_{n\to\infty}\int_0^1 \! f(x^n) ...
1
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1answer
27 views

What can we say about open unit balls of sup-norm and integral-norm

Consider the normed linear spaces $X_1=(C[0,1], ||.||_1)$ and $X_{\infty}=(C[0,1],||.||_{\infty})$ , where $C[0,1]$ denotes the vector space of all continuous real valued functions on $[0,1]$ and ...
1
vote
1answer
25 views

$P$ is a monic polynomial of degree $n$ , then which are correct?

Suppose that $P$ is a monic polynomial of degree $n$ in one variable with real coefficients and $K$ is a real number. Then which of the following statements are necessarily correct ? If $n$ ...
0
votes
1answer
43 views

How Can I Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+.. $ [duplicate]

Proof $1+2\cos(\theta)+2\cos(2\theta)+2\cos(3\theta)+...+2\cos((n-1)\theta)=\frac{2\sin((n-\frac{1}{2})\theta)}{2\sin(\frac{1}{2}\theta)} $
1
vote
1answer
20 views

Finding a differentiable function from (0,1) to (0,1) dominating (point wise) a given continuous function from (0,1) to (0,1)

Suppose we have a continuous function $f: (0,1) \to (0,1)$. Does there exist a differentiable function $\phi: (0,1) \to (0,1)$ such that $f(x) \leq \phi(x)$? There does exist such a differentiable ...
0
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2answers
76 views

Example of a set $S$ that is countable, but the set of limit points is uncountable [on hold]

What would be an example of a set $S$ so that $S$ is countable. However $S'$ is uncountable. In this $S'$ is the set of all the limit points of $S$.
7
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4answers
124 views

If $\int_0^{x/3} f(t)dt =\int_0^xf(t)dt$, prove $f$ is identically $0$

$f:[0,1] \to \mathbf R$ is continuous. If $$\int_0^{x/3} f(t)dt =\int_0^xf(t)dt$$ for all $x$ in $[0,1]$, prove that $f$ is identically $0$. My thought is to prove that the maximum and minimum of ...
1
vote
1answer
34 views

What does bounded partial derivatives exactly mean?

This might be a naive question, but if I give myself a continuously differentiable function $f$ from $\mathbb{R}^n$ to $\mathbb{R}$ which is said to have bounded partial derivatives, does this mean ...
2
votes
2answers
43 views

convergence proof without finding 'N'

I tried to prove $\sqrt{n^2 +n}-n$ converges to $\frac{1}{2}$ I am not sure what I have proved is correct. $$\left|\frac{n}{\sqrt{n^2+n} +n} - \frac{1}{2} \right| = \left| \frac{2n - ...
0
votes
0answers
27 views

Banach contraction mapping to find unique solution of $y(x) = \int_0^1 G(x,s) f(s,y(s)) ds$

Problem statement: Use the Banach contraction mapping theorem to show that $$y(x) = \int_0^1 G(x,s)\ f(s,y(s))\ ds$$ has a unique, continuous solution $y = y(x)$ if (1) $G(x,s) \ge 0$ for $0 \le x, ...
2
votes
1answer
44 views

Derivative exists by first principles but undefined when using chain rule

Consider the function $h$ defined by \begin{align} h(z,y)=(z^3+y^3)^{\frac{1}{3}} \end{align} Then \begin{align*} h_z(0,0)&=\lim_{t\rightarrow 0}\frac{(t^3)^{\frac{1}{3}}}{t}\\ &=1 ...
-1
votes
1answer
41 views

Convergence of $E_n=(-n,n]$ [closed]

If $E_n=(-n,n]$ is a sequence of sets, does it converge to $(- \infty , \infty)$ or $(- \infty , \infty ]$ or $[- \infty , \infty ]$? What is the proof?
2
votes
1answer
19 views

$F$ smooth $\implies F=p=q$.

This is a follow-up from my earlier question: Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, then $p=q$. Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be ...
4
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0answers
85 views
+100

Solving ODE rigorously

I am given the ODE $$(f''(r)+\frac{f'(r)}{r})(1+f'(r)^2)-f'(r)^2f''(r)=0$$ and want to solve it rigorously for $r>0.$ So especially, I don't want to loose any solutions. $\textbf{Derivation of ...
1
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2answers
40 views

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, then $p=q$.

Let $p,q$ be real polynomials. Let $F: \Bbb R \to \Bbb R$ be differentiable, defined by: $$ F = \begin{cases} \hfill q \hfill & \text{X $\geq$ a} \\ \hfill p \hfill & ...
2
votes
0answers
101 views

Higher infinities without Set Theory

Apart from Cantor's diagonalization argument, there are a number of ways to show that cardinality of R is greater than that of N (eg: Baire Category theorem, path connectedness of R and so on). Are ...
1
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0answers
37 views

Generating structure of Borel field

On P.32 of [P.Billingsley] Probability and Measure, 3ed, 1993, the author wrote: ...and there are Borel sets that cannot be arrived at from the intervals by any finite sequence of set-theoretic ...
1
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1answer
36 views

Differentiable not $C^1$ and Darboux property

Are there any differentiable not $C^1$ function $f: [0,1] \to \mathbb{R}$, $f'(0)<f'(1)$ such that there exists $c \in (f'(0),f'(1))$ that $f'$ doesn't reach value $c$? Classical example of ...
1
vote
1answer
44 views

Absolute Value Property of Field of Real Numbers

I don't think my thought process is correct. Also, does 'if and only if' indicate that I should automatically resort to proof by contradiction? Show that ${|b|} \le {a}$ if and only if $ {-a} \le b ...
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0answers
21 views

Derivatives of ordinal order

This question actually arises from this answer to another question, which contains the sentence A function is smooth is it has derivatives of infinite order. While the author surely didn't ...
2
votes
1answer
22 views

Finding a convex function dominated (point wise) by a given positive function on (0,1)

This problem is actually the Exercise 8 in Chapter 3 of Rudin's Real and Complex analysis book. The problem is as follows: If $g$ is a positive function on $(0,1)$ such that $g(x) \to \infty$ as $x ...
2
votes
2answers
51 views

Show that the following mapping is a contraction.

I have the following problem from a past paper: "Show that the mapping, $$T(x_1,x_2)=\left(\frac{x_1+2x_2}5-1,\frac{x_1-2x_2}7+1\right)$$ is a contraction on $(\mathbb R^2,d_\infty)$." I ...
0
votes
1answer
35 views

$a_n\leqslant c, a_0=c$, then $\limsup_n a_n\leqslant c$?

(1.) If I know that $a_n\leqslant c, a_0=c, a_n<c, n\geqslant 1$, does then follow that $\limsup_{n\to\infty}a_n\leqslant c$? (2.) If I have a sequence $(b_n)$ and I if I want to show that ...
1
vote
1answer
53 views

Limit of arithmetic means

If $\{s_n\}$ is a complex sequence, define its arithmetic means $\sigma_n$ by $$\sigma_n=\dfrac{s_0+s_1+\dots+s_n}{n+1}.$$ If $\lim s_n=s$, prove that $\lim \sigma_n=s.$ My proof: Let $t_n:=s_n-s$ ...
1
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0answers
17 views

Higher order terms in Taylor expansion tend to infinity faster.

Suppose $g$ is a smooth bounded and symmetric probability density function (pdf). Let $\{(X_1,Y_1), ..., (X_N,Y_N)\}$ be a random sample from the joint pdf $t(x,y)$. Further assume $a\to 0$ and $Na ...
4
votes
3answers
73 views

Sum of $\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$

I've been working with the series: $$\sum_{n=1}^{\infty}(-1)^{n+1}\frac{x^{2n-1}}{2n-1}$$ From the ratio test it is clear that the series converges for $|x| < 1$, but I'm unable to obtain the sum ...
0
votes
1answer
37 views

Continuity of integral whose domain of integration depends continuously on time

Let $\Omega(t)$ be a region in $\mathbb{R}^n$ that depends continuously on some parameter $t$. Let $f$ be a function defined on $\mathbb{R}^n$ such that $f > 0$ almost everywhere. Then, can we ...
1
vote
1answer
29 views

Completeness Axiom Proof

Let $A_1$, $A_2$, $A_3$... be a collection of nonempty sets, each of which is bounded above. (a.) Find a formula for sup($A_1$$\bigcup$$A_2$). Extend this to sup($\bigcup^{n}_{k=1}$$A_k$). (b.) ...
0
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0answers
79 views

If $f^{-1}(x)$ is continuous, is $f(x)$ also continuous?

Let $f:\mathbb{R}\mapsto\mathbb{R}$ be a one-to-one function with $f(\mathbb{R})=\mathbb{R}$. If $f^{-1}(x)$ is continuous $\forall x\in\mathbb{R}$, prove or disprove that $f(x)$ is continuous ...
-3
votes
0answers
72 views

$\int \left(\cos\left(e^{\sin(x)}\right)\right) \text{d}x$ [closed]

I have to find this integral but I have no idea if there is any solution to it: $$\int \left(\cos\left(e^{\sin(x)}\right)\right) \text{d}x$$
3
votes
0answers
29 views

Zero Gaussian curvature and restriction estimates

Let $$ \left(\int_M|\hat{f}|^qd\mu(\xi)\right)^{1/q}\leq c||f||_{L^p(\mathbb{R}^n)} $$ be a restriction estimate for a hypersurface $M\subset\mathbb{R}^n,~1<p<\infty$ and $\mu$ a ...
1
vote
1answer
36 views

Gradient flow of Dirichlet energy

I have heard that the gradient flow of the Dirichlet energy gives a solution of the heat equation, i.e. if $u(t,x) \in C^1( [0,\infty) \times \mathbb R^d)$ solves $$ u_t(t,x) = - dE(u(t,x)), $$ where ...
0
votes
1answer
35 views

L'Hôpital with absolute continuity

I have been studying for my real analysis qualifying exam, and I have noticed a trend of questions similar to the following: Suppose that $f$ is absolutely continuous, $f'\in L^3$, and $f(0)=0$. ...
1
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1answer
49 views

Convergence of $\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$

I have trouble with the following sum: $$S=\sum_{n=1}^{\infty}(-1)^n\frac{x^n+1}{n}$$ Since $a_n=\frac{2}{n}$ decreases monotonically and tends to $0$, it converges by Liebniz criterion. Then, by ...
3
votes
2answers
140 views

Proving that $\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos (\tan (x))-\cot (x))\cot (x) \, dx=\frac{\pi(e-2)}{2e}$

I think one of the ways of doing it is by the use of the differentiation with parameter. Do you see an easy way of calculating it by real methods? $$\int_0^{\pi/2} (\sin (\tan (x))+\cot (x) \cos ...
-4
votes
1answer
43 views

If a sequence of real numbers converges to 1, then what do we say about its reciprocal?

Suppose the sequence of real number $\lbrace x_n \rbrace _{n=1} ^{\infty}$ converges to 1, $x_n \neq 0 \; \; \; \forall n=1,2,3,...$. Show that $\lbrace \frac{1}{x_n} \rbrace _{n=1} ^{\infty} $ ...
4
votes
2answers
69 views

Convergence of $\lbrace \sin( \frac{1}{n}) \rbrace $ [closed]

I believe that the sequence $\lbrace \sin( \frac{1}{n}) \rbrace $ converges to zero. Can someone give an episilon-delta proof of this fact?
1
vote
1answer
34 views

prove that $\displaystyle \lim_{t \to a} g_1(t)f_1(t) + g_2(t)f_2(t) = \lim_{t \to a}(g_1(t)+g_2(t))f_1(t)$

$a \in R$ , If $$\displaystyle \lim_{t \to a}\left( g_1(t) + g_2(t)\right)~~exist, ~~when ~~g_1(t)+g_2(t) > 0$$ and $$\displaystyle \left(\lim_{t \to a} f_1(t)\right) = \left(\lim_{t \to ...
3
votes
2answers
65 views

How to calculate $\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$

How can I compute the following limit? $$\displaystyle \lim_{\alpha \to -1}\left(2\Gamma(-\alpha-1)+\Gamma\left(\frac{\alpha + 1}{2}\right) \right)$$ The answer appears to be about -1.73 I would be ...
2
votes
0answers
37 views

On the common zeros of $1-x\tan{x}$ and $1-\frac1x \arctan{\frac1x}$

This is ON HOLD on MO. Let $f(x)=1-x\tan{x}$. Let $g(x)=1-\frac1x \arctan{\frac1x}$. Let $r=0.8603335890193797624838\ldots$ be a real root of $f(x)=0$. High precision numerical computations ...
1
vote
2answers
61 views

Is a particular countable subset of the Cantor set Polish?

Consider the Cantor space $\mathcal{C} := \{ 0, 1 \}^{\mathbb{N}}$ and the subset $\mathcal{T} \subseteq \mathcal{C}$ of sequences that start with $1$ and eventually "terminate" with $0$, i.e. ...
1
vote
2answers
40 views

To show that $f(x)= \frac{1}{x}$ is not uniformly continuous on $(0,\infty)$

To show that $f(x)= \frac{1}{x}$ is not uniformly continuous on $(0,\infty)$ ATTEMPT Choose $\epsilon$ to be less than 1 Let $x_1=\delta$ and $x_2=\frac{\delta}{2}$ Then $|x_1-x_2|$ is less than ...
0
votes
3answers
29 views

Power series with radius convergence $\leqslant 1$

Suppose that the coefficients of the power series $\sum a_nz^n$ are integers, infinitely many of which are distinct from zero. Prove that the radius of convergence is at most 1. Proof: Let radius of ...
0
votes
1answer
8 views

A point $a=(a_1,…,a_n)$ is isolated point in the cartesian product

Let be $M$ a metric space. A point $a=(a_1,...,a_n)$ is isolated point in the cartesian product $M=M_1\times...\times M_n$, if and only if, each coordinates $a_i$ is a isolated point in $M_i$ My ...
0
votes
0answers
17 views

surface and Lebesgue measure

I'm trying to proof that surface measure $\sigma_m$ on manifold $M$ and Lebesgue measure $\lambda_m$ coincides on Borel sets. I know that I have to use that locally, $M$ is very well approximated by ...
2
votes
1answer
54 views

What is the smallest subfield of the complex numbers which has the property that every polynomial of odd degree has a root

It can be shown using the intermediate value theorem that every polynomial of odd degree with real coefficients must have at least one real root. I was just curious, are there any other smaller fields ...
2
votes
1answer
25 views

$ 1+\mathsf{d}^\star(B) \le \mathsf{d}^\star(A\cup B)+\mathsf{d}^\star(B\cup C) $ with $A\cup B\cup C=\bf{N}^+$

For each subset of positive integers $X\subseteq \bf N^+$ define the upper asymptotic density as $$ \mathsf{d}^\star(X)=\limsup_{n\to \infty} \frac{|X\cap [1,n]|}{n}. $$ Problem: Let $A,B,C$ be a ...