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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Convergence in measure of product of convergent sequences

Let $(X,\Sigma,\mu)$ be a finite measurable space ($\mu(X)<\infty$). Suppose $f_n \xrightarrow{\mu} f$ and $g_n \xrightarrow{\mu} f$, prove that $f_ng_n \xrightarrow{\mu} fg$ I'll write what I ...
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If f is continuous and strictly increasing, then the function $f^{-1}:f(I)\rightarrow I$ is continuous and strictly increasing.

Let $I \subseteq \Bbb R$ be a non-degenerate open interval, and let $f:I\rightarrow \Bbb R$ be a function. Suppose that f is strictly monotone. If f is continuous and strictly increasing (or ...
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1answer
11 views

Real analysis: simple second order ODE

I'm studying real analysis at the moment (just covered the mean value theorem, constancy theorem, applications to DEs etc.) and have run across this question that I'm stuck on. Any help would be much ...
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3answers
20 views

Show that the closed ball is closed in $\mathbb{R}^p$

Let $r>0, p \in \mathbb{N}$ be given. Show in detail that the closed ball $\{ x \in \mathbb{R}^p : ||x|| \leq r \}$ is closed in $\mathbb{R}^p$. Let $A = \{ x \in \mathbb{R}^p : ||x|| \leq r ...
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1answer
29 views

How to express sum as triple summation

I am trying to express the following sequences as summations: $$ 1+2^2+3^2+4^4+5^4+6^4+7^4 $$ and $$ 1+(2+3)^2 + (4+5+6+7)^4 $$ as summations. I think they will likely be triple summations, so ...
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13 views

base b expansion of real numbers

This is a problem in Zygmund's analysis book. It is intuitively very straightforward. However, I could not give a rigorous proof. I hope someone can show me how to prove this rigorously. Problem: ...
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55 views

Evaluate $\lim _{n\to \infty }\int_1^2\:\frac{x^n}{x^n+1}dx$

We have $$I_n=\int _1^2\:\frac{x^n}{x^n+1}dx$$ and we need to find $\lim _{n\to \infty }I_n$. Have any ideea how we can evaluate this limit?
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1answer
34 views

What are the fixed points of $f_ c = c · \sin$ for $c > 1$?

I’m doing an exercise for a lecture on dynamical systems. We are asked to classify all bifurcations of the dynamical system $f_c = c·\sin$ for real $c > 0$. We are given that bifurcations of ...
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How to prove an extremum existence in problems, regarding calculus of variations

Let's consider a functional $S(y)=\int_{a}^{b}{f(x, y, y') \cdot dx}$. It's known that if the function that attains minumum or maximum to $y(x)$ does exists, then it can be got from the Euler-Lagrange ...
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27 views

showing a function is continuous

While working on analysis continuity, i came across this question.I have understood the concept of continuity but cant see how to answer this question by including epsilon delta. $$f(x) ...
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36 views

Real Analysis book with pictures and ideas of proofs

I am taking real analysis course in my graduate class of Maths. My classes will start in 3 months. I have studied real analysis but not very rigorously. Whenever I see theorem I have no idea on how ...
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20 views

Compute the values of two infinite products whose factors are the same

I have the following question: How to prove that $(1-\frac{1}{2})\cdot (1+\frac{1}{3})\cdot (1-\frac{1}{4})\cdot (1+\frac{1}{5})\cdot (1-\frac{1}{6})\cdot (1+\frac{1}{7})\cdot ...
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4 views

Prove the Lagrangian reminder term $lim_{x\rightarrow x_0}\frac{\xi_n-x_0}{x-x_0}=\frac{1}{n+2}.$

For Taylor expansion $$f(x)=f(x_0)+f^{'}(x_0)(x-x_0)+\cdots+\frac{f^{(n+1)}(\xi_n)}{(n+1)!}(x-x_0)^{n+1},$$ if $f^{(n+2)}(x_0)\neq 0,$ how to prove $$lim_{x\rightarrow ...
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1answer
25 views

To show that a function defined by integral is absolutely continuous

Let $$ F(x)=\int_{[0,x]\times[0,x]}f,\quad x\in[0,1] $$ Here f is a Lesbegue-integrable on the unit square $[0,1]\times[0,1]$. I need to show that $F$ is absolutely continuous and express the ...
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53 views

As the limit of $n$ goes to infinity, prove that $x^n = 0$ if $\operatorname{abs}(x)<1$.

As the limit of $n$ goes to infinity, prove that $x^n = 0$ if $\operatorname{abs}(x)<1$. So I want to prove it this by observing that $\operatorname{abs}(x) < 1$ which means ...
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18 views

Sequence of measurable functions on a null set

I am trying to do this problem: Let $(X,\Sigma,\mu)$ be a measurable space and let $E \in \Sigma$. Let $(f_k)_{k \in \mathbb N}:E \to \mathbb R$ be a sequence of measurable functions such that for ...
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13 views

Integral inequality of a continuous function on a compact set

Let $\Omega$ be a compact subset of $\mathbb{R}^n$, and let $f:\Omega \to \mathbb{R}$ be a continuous positive function. Let $V_1$ and $V_2$ be subsets of $\Omega$. Then, I like to show that there ...
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3answers
35 views

Proving uncountability of $\mathbb R$ only using the complete ordered field axioms

If we define the real numbers abstractly as a complete ordered field (like described in the Wikipedia page), how can we prove that they are uncountable? In other words, using just the axioms of a ...
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1answer
33 views

Show that a function is continuous iff it is constant

show that a function from $\mathbb{R}$ with the standard metric to $\mathbb{R}$ with the discrete metric is continuous if and only if it is constant. The solution states to use the $\epsilon, ...
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16 views

Stieltjes Integral - If $f, f^2, g, g^2\in R(\alpha)$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$

My question is if $f, f^2, g, g^2\in R(\alpha)$ on $[a,b]$ for an arbitrary integrator $\alpha$, then is $fg\in R(\alpha)$ as well? This question stemmed from a problem in Apostol's Analysis, in ...
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27 views

Partial ordering of functions

Let $X$ be the set of all real-valued functions $x$ on the interval $[0,1]$ and let $x \leq y$ mean that $x(t) \leq y(t)$ for all $t \in [0,1]$. Does it define a partial ordering/ total ordering? Does ...
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37 views

How can I prove fundamental theorem in measure theory

How can I prove that $1)$ if $f$ and $g$ are measurable functions such that $0 \le f \le g$ Then $\int_{X}{}fdμ \le \int_{X}{}gdμ$ $2)$ if $f$ and $g$ are integrable functions such that $f \le g$ ...
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1answer
8 views

Show that if $U$ is open , then $ Fr ( p(\overline U) ) \subset p(\overline U ) \cap p( X - U)$

Question : Let $ P : X \rightarrow Y$ be a closed surjection, . If $U$ is open , then $ Fr ( p(\overline U) ) \subset p(\overline U ) \cap p( X - U)$ . I know that $ Fr (p(\overline U ...
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1answer
18 views

A proof related to diameter of a simplex S

Question: Prove that the diameter $\mathcal p(S)$ of a simplex $\mathcal S$ equals the greatest Eucledian distance between two vectors in the simplex. My opinion: We all know what every vector in the ...
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1answer
24 views

$L^{1} \cap L^{\infty}$ is dense is in $L^{p}$

Why is $L^{1} \cap L^{\infty}$ dense is in $L^{p}$? my work: i use the $$\Vert f \Vert_{p} \leq \Vert f \Vert^{1/p}_{1} \Vert f \Vert_{\infty}^{1-1/p}$$
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21 views

constructing an example of a ball of larger radius, contained in a ball of smaller radius

I've found an example where there is a ball of larger radius contained in a ball of smaller radius, but I'm not sure how it works: Let $X = \{ x \in \mathbb{R}^2 : x_1^2 + x_2^2 \leq 9 \}$ with the ...
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32 views

Computing limits using Monotone Convergence theorem

I am trying to compute the limits of $\lim_{n \rightarrow \infty} \int\limits_0^{\infty} \dfrac{1}{(1+\dfrac{x}{n})^n \sqrt[n]x}dx $ by using Monotone convergence theorem of integrals and switching ...
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24 views

$d_1(x,y)=|x-y|$ & $d_2(x,y)=|\arctan(x)-\arctan(y)|$ equivalent on $\mathbb R$?

We call two metrices equivalent if for all sequences $x_n,y_n\in\mathbb R$ it holds $\lim_{n\to\infty}d_1(x_n,y_n)=0 \iff\lim_{n\to\infty}d_2(x_n,y_n)=0$ . I have given $d_1(x,y)=|x-y|$ and ...
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Show that any subset of $(\mathbb{N},d)$ is open and closed

Show that any subset of $(\mathbb{N},d)$ is open and closed, where $$d(m,n) = \frac{|m-n|}{1+|m-n|}$$ my attempt: let $A \subset \mathbb{N}$ then for any $x \in A$ we have that $B(x,1/3) = \{x\} ...
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16 views

Integrating 2 form over torus

Let $\Bbb M^2 ⊂ \Bbb R^3$ be the torus of revolution obtained by rotating the circle $(x−2)^2 +z^2 = 1$ in the $xz$ plane around the $z$ axis. Consider the orientation on $M$ induced by the ...
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Existence of real valued function continuous at $\mathbb Q$ discontinuous at $\mathbb R\backslash \mathbb Q$ [duplicate]

Does there exist a real-valued function of a real variable which is continuous at every rational point and discontinuous at every irrational point?
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If $x \in (b, \infty)$ Show there exists a natural number $n_0$ such that $x > b + \frac{1}{n_0}$

Assume $\displaystyle \lim_{n \to \infty} \frac{1}{n} =0$ and for $b \in \mathbb{R}$, $\displaystyle \lim_{n \to \infty} b = b$. WITHOUT using the Archimedean postulate, show that if $x \in (b, ...
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47 views

$\int_a^b f(x) g'(x) dx = 0$ implies $f$ is constant

Given $f$ is continuous on $[a,b]$, $\forall g$ which is a continuously differentiable function on $[a,b]$, with $g(a)=g(b)=0$, the following equation is satisfied: $\int_a^b f(x) g'(x) dx = 0$. I ...
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57 views

Show that the antiderivative exist [on hold]

I am new to this. How do I show that the antiderivative exist and show that is continuous too? Thanks
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1answer
51 views

Explanation for absolute value

So $f_a:R\rightarrow \:R,\:f_a(x)=\:\frac{1}{\left|x-a\right|+3}$, and we have to evaluate $\lim _{a\to \infty }\int _0^3\:f_a\left(x\right)dx$. But $\left|x-a\right|\:$ is equal with: ...
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15 views

Question on Limits; Invariance Principle

This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. "E1. Starting with a point S (a, b) of the plane with 0 < b < a, we generate a ...
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3answers
41 views

Limit function of $\frac{\epsilon}{\epsilon^2+x^2}$ as $\epsilon\to 0$

What is the limit of the sequence of functions$\frac{\epsilon}{\epsilon^2+x^2}$ as $\epsilon\to 0$? I think this just doesn't exist, since it goes to $\infty$ in $x=0$ and goes to $0$ everywhere ...
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23 views

Riemann integral property proof using the definition

We say that a function $f:[a,b]\to \mathbb{R}$ is Riemann integrable if for every $\epsilon>0$, there are two step functions $g_1,g_2$ such that $g_1 \leq f \leq g_2$ and $\int_a^b ...
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If $d(x_0,y_j)\to d(x_0,y_0)$, then $y_j \to y_0$.

Consider a metric space $X$, and a compact subset $C\subset X$.Let $x_0\in X-C$. We can show that there is a point $y_0\in C$ such that $d(x_0,y)=\inf_{y\in C} d(x_0,y)$. Now suppose there is ...
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1answer
20 views

given the following two conditions, find $f(x,y)$

Suppose that a function $f$ defined on $\mathbb R^2$ satisfies the following conditions: $f(x+t,y)=f(x,y)+ty$; $f(x,t+y)=f(x,y)+tx$; $f(0,0)=k$; then for all $x,y \in\mathbb R$, $f(x,y)=$ a) ...
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37 views

Convex set of derivatives implies mean value theorem

Let U$ \subset$ $R^{^{n}}\ $be open, $f:U\rightarrow R^{m}$ differentiable on U, and segment $[a,b]\subset U$. Assume that the set of derivatives $\{ f'(x)\in L(R^{^{n}},R^{^{m}}):x\in [a,b] \}$ ...
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1answer
16 views

Extending the definition of curve length

I know for continuously differentiable curves on closed interval $[a,b]$, the curve length is given by $\Lambda (\gamma)=\int_a^b |\gamma^{'}(t)|dt$. But what about curves such that $\gamma^{'}(t)$ is ...
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(soft question) why do we study complex measures and complex-valued functionals in modern analysis

Recently I am struggling with "complex" things for my "real" analysis class. We are using Folland's Real analysis, 2nd for text book. It seems that Folland is trying to use complex-valued functions ...
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58 views

If the derivative is $0$, then $f$ is constant in a banach space

My question is simple. Take a differentiable function $f: U \subset \mathbb{E} \rightarrow \mathbb{F}$, where $\mathbb{E}, \mathbb{F}$ are banach spaces and $U$ is an open connected subset of ...
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1answer
14 views

Is this function differentiable w.r.t. a variable in an indicator?

I have $$y^i = x^i - \alpha \sum_{j \epsilon N} (x^j - x^i) I_{x^i \lt x^j} - \beta \sum_{j \epsilon N} (x^i - x^j) I_{x^i \ge x^j}$$ where N = {1, 2, ..., n}, and $I_{x^i \lt x^j}$ is 1 when $x^i ...
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1answer
29 views

Integral relations when using different measures.

Let $(X,\mathcal{M})$ a measurable space and $\mu$,$\nu$ two non-negative measures s.t $\mu \geq \nu$. Does it hold that $\int_E f \, d\mu \geq \int_E f \,d\nu $ where $E \in \mathcal{M}$. I suspect ...
5
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1answer
32 views

If $E \subset\mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E) \in E$

Suppose that $E \subset \mathbb R$ is compact and nonempty. Prove $\sup (E)$, $\inf (E)\in E$. attempt: Suppose $E$ is compact, then $E$ is closed and bounded. Thus $\sup(E)$ and $\inf (E)$ exist. ...
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1answer
21 views

If a continously differentiable function has a local minimizer, can it be one to one?

Let $f$ be a continuously differentiable function defined $f : \mathbb R \to \mathbb R$ such that $f(x)$ is defined for for all $x$. Suppose $x_0$ is a local minimizer for $f$. Is $f$ one-to-one? I ...
2
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1answer
38 views

Convergence of Taylor Series

Prove that if $f$ is defined for $|x|< r$ and if there exists a constant $B$ such that $$| f^n(x) |\le B$$ for all $|x|< r$ and $n \in \mathbb N$, then the Taylor series expansion : ...
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2answers
49 views

Differentiability and continuity at the origin of piecewise defined $g(x,y) = y-x^2$, $y+x^2$, or $0$

$$g(x,y)= \begin{cases} y-x^2, & y\ge x^2\\ y+x^2, & y\le -x^2\\ 0 & \text-x^2\le y\le x^2 \end{cases}$$ I need to find all the directional derivatives at the origin in the tangent ...