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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2
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2answers
52 views

Power series of trigonometric functions

Problem statement: Determine those $x$, for which the power series is convergent and determine the sum. $$f(x)=x+\sum_{n=2}^{\infty}(-1)^{n-1}2n\frac{x^{2n-1}}{(2n-1)!}$$. Progress: I have ...
0
votes
2answers
164 views

Bounded, measurable and supported on a set of finite measure function

Suppose f is a bounded and measurable function on R and supported on a set of finite measure. Prove that for every $\epsilon \gt 0$ there exists a simple function $s$ such that $\int |f-s|dx$ $\lt ...
1
vote
2answers
41 views

Sequence limits

Let $x_n \in \mathbb{Z}$, $x \in \mathbb{R}$ and $lim(x_n)=x$. Show from first principles that $x_n$ is eventually equal to $x$. I started by just defining the lim. For $\varepsilon>0$, $\exists N ...
0
votes
1answer
81 views

lim sup of two sequences

Let $(a_n)_{n \in\ \mathbb{N}}$ a bounded sequence in $\mathbb{R}$. For $n \in \mathbb{N}$ let $$v_n=\sup\{a_k; ~k \geq n\},\quad u_n=\inf\{a_k; ~k \geq n\},\quad s_n=\sup\{|a_k-a_l|; ~k,l \geq ...
0
votes
1answer
60 views

integral partition, real analysis

I'm struggling with this question: If we have $f(x) = x^2 $ and $P_n $ which partitions $[1,3]$ into $n$ sub-intervals, each equal in length,how can I write the formulas for $L(f,P_n)$ and $U(f,P_n)$ ...
1
vote
1answer
78 views

What can we say about the intersection of a clopen set and a connected set

Let $X$ be a metric space and let $E \subseteq X$ be a clopen subset. let $A \subseteq X$ be a connected subset. What can be said about $A \cap E$ when $A \cap E \neq \varnothing$? I believe that $E ...
1
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2answers
60 views

Assume $X$ and $Y$ be nonempty subsets of $R$ such that $x<y$ for every $x \in X$ and $y \in Y$. Prove that $\sup X \leq \inf Y$. Is my proof correct?

We have that both sets are nonempty, by the completeness axiom, both sup and inf exist for both sets. Since $x < y$, $y$ is an upper bound for $X$ and hence by def. of sup, sup$X \le y$. So $\sup ...
2
votes
0answers
143 views

Each bounded measurable function $f:[a,b]\to\mathbb{R}$ is almost a Borel function.

The exact problem statement is: There exists a Borel function $h:[a,b]\to\mathbb{R}$ and a Borel set $H\subset[a,b]$ such that $f=h$ on $H$ and $m([a,b]\setminus H)=0$. My attempt of the proof is as ...
0
votes
1answer
44 views

Convergence of an analytic function

Let $f:\mathbb{R} \rightarrow \mathbb{R}$ be a smooth function. Let $R$ be the radius of convergence of the Taylor series centered at $a.$ For each $n \in \mathbb{N},$ let $M_n= \sup\{f^{n}(t) : t \in ...
0
votes
1answer
92 views

epsilon-dense property

we were learning about epsilon-dense sets: If $S$ is a subset of a metric space, then $T$ in $S$ is $\epsilon$-dense in $S$ for given $\epsilon$ if for any $s$ in $S$, there is $t$ in $T$ s.t. the ...
6
votes
2answers
101 views

Convergence of a certain series.

If $(q_n)$ is an enumeration of rationals in $(-1,1)$ except $0$, is the following series convergent? $$ \sum_{n=1}^{\infty}\frac{1}{(nq_n)^2} $$ Are there enumerations for which is convergent and ...
1
vote
1answer
36 views

confused with the definition of the limit point

Let p be a limit point of k. by definition of limit point, for each n$\in N$ there exist $P_n \in K $ with $P_n \ne p $ such that $$|p_n - p| < 1/n $$ I do not understand the part 1/n. why they ...
5
votes
2answers
226 views

Is there a proof of the irrationality of $\sqrt{2}$ that involves modular arithmetic?

I was reading Ian Stewart's Concepts of Modern Mathematics. Using congruences, It's possible to explain why all perfect squares end in $0,1,4,5,6,9$ but not in $2,3,7,8$. With this I had the ...
1
vote
0answers
35 views

open cover for one point concept

Is it possible that One point can be called an open cover? can you explain the detail about this concept? and if you have example that will be greate thank you
2
votes
1answer
75 views

$F(xy) = F(x)+F(y)$ Proof

Suppose $F$ is differentiable $\forall x>0$ and $F(xy) = F(x)+F(y) \forall x,y>0$. Prove that if $F$ is not the zero function, then $\exists a>0$ such that $F(x)=log_a(x)\forall x>0$. I ...
1
vote
2answers
63 views

Evaluate $\lim_{n\to\infty} n^2(1-\cos\frac{x}{n})$

Find this limit by utilizing Lagrange Remainder Theorem: $$\lim_{n\to\infty} n^2\left(1-\cos\frac{x}{n} \right)$$ I have tried using the definition and can't get it.
3
votes
1answer
610 views

Heisenberg uncertainty principle in $d$ dimensions.

Suppose $f(x)$ is a $d$-dimensional real function and $\int_{R^{d}}|f(x)|^2dx=1$. Show that $$ (\int_{R^{d}}|x|^2|f(x)|^2dx)(\int_{R^{d}}|\xi|^2|\hat f(\xi)|^2d\xi)\geq\frac{d^2}{16\pi^2}$$ I ...
0
votes
1answer
32 views

help with Lipschitz constant

Find the Lipschitz constant of $$ g(x)=\sqrt{x} $$ for $$1\leqslant x\leqslant2$$ please any idea how to start?
3
votes
3answers
177 views

Convergence of $\sum\limits_{n=0}^{\infty}\frac{n-\sqrt{n}}{(n+\sqrt{n})^2}$

I want to check, whether $\sum\limits_{n=0}^{\infty}\frac{n-\sqrt{n}}{(n+\sqrt{n})^2}$ converges or diverge. I tried to use the Ratio test: \begin{align} \left|\dfrac{a_{n+1}}{a_n}\right| & = ...
5
votes
2answers
169 views

Writing function as infinite Fourier sum with sine kernel

Let $f\in L^2(\mathbb{R})$ be such that $\hat{f}$ is supported on $[-\pi,\pi]$. Also, $f$ is continuous and goes to $0$ at $\pm \infty$. Let $K(y)=\dfrac{1}{\pi y}\sin(\pi y)$. Show that ...
2
votes
2answers
131 views

For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ what does $\lim_{x\rightarrow +\infty} f'(x)=1$ imply? (TIFR GS $2014$)

Question is : For a differentiable function $f:\mathbb{R} \rightarrow \mathbb{R}$ what does $\lim_{x\rightarrow +\infty} f'(x)=1$ imply? Options: $f$ is bounded $f$ is increasing $f$ is unbounded ...
0
votes
0answers
65 views

Show that this function is a diffeomorphism

Define $\Omega:=\mathbb{R}^n\setminus\overline{B_R}(0)$ with $R>0$ and $n>1$ and $G:=B_R(0)\setminus\left\{0\right\}$ Show that the function $$ \phi\colon \Omega\to G, ...
2
votes
1answer
36 views

Why this convergence should be uniform?

Let $f$ be an extended real valued measurable function. Then we need to show that there exists a sequence of real-valued simple functions that converge to $f$. We also need to show that if $f$ is ...
26
votes
4answers
4k views

Is every Lebesgue measurable function on $\mathbb{R}$ the pointwise limit of continuous functions?

I know that if $f$ is a Lebesgue measurable function on $[a,b]$ then there exists a continuous function $g$ such that $|f(x)-g(x)|< \epsilon$ for all $x\in [a,b]\setminus P$ where the measure of ...
0
votes
1answer
230 views

Bounded measurable function

Suppose $f$ is a bounded measurable function and $m\{x\in \mathbb{R}:f(x)\ne 0 \} > 0$. Prove that for every $\epsilon>0$ there exists a simple function $s$ such that $$\displaystyle \int ...
0
votes
2answers
55 views

question about real analysis concerning inequality

Let $\epsilon > 0$ be given. Suppose we have that $$a - \epsilon < F(x) < a + \epsilon$$ Does it follow that $a - \epsilon < F(x) \leq a $ ??
0
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1answer
40 views

Clarification of a set description needed

S = {x in M | d(x,a) < d(x,b)} for any a,b in a metric space (I am not sure what this set is). Is S closed or open?
4
votes
1answer
164 views

Some Scaling Estimate for Heat Kernel

NOTE. I have rewritten the question to summarize my current progress on this question. The bounty is for completing what I have done so far, or by offering a more elegant solution probably based on ...
0
votes
2answers
58 views

Elementary question about real analysis

Suppose $A \subseteq \mathbb{R}$, and let $x \in [a,b]$. Put $$ F(x) = \int_{[a,x]} \chi_A dm $$ Then of course we know that $F $ is differentiable for almost all $x \in [a.b]$ and $$ F'(x) = 1 \; ...
0
votes
1answer
38 views

Show that $(C,\rho)$ is separable.

Suppose $C$ is the class of all real-valued continuous functions on $[0,1]$. Define the metric on $C$ to be $\rho(f,g)=\sup\{|f(t)-g(t)|\}$, for $f,g\in C$. We need to prove that $(C,\rho)$ is a ...
0
votes
3answers
722 views

Compact sets of metric spaces are closed?

I am struggling with the idea that all compact subsets of a metric space are closed after reading chapter 2 of Rudin's Principles of Mathematical Analysis. The reason I am confused is that it seems ...
0
votes
1answer
61 views

Metric Space Distance Question

for metrics $d_1$ and $d_2$ on a nonempty set $X$, suppose there are positive constants $a$ and $b$ such that $ad_1(x,y) \le d_2(x,y) \le bd_1(x,y), \forall x,y \in X$. Prove that a subset $U$ of $X$ ...
0
votes
1answer
46 views

Push-forward vector field for constant vector field

Let $v$ be a constant vector field $v=\sum_{i=1}^n c_i\dfrac{\partial}{\partial x_i}$, and let $f:\mathbb{R}^n\rightarrow\mathbb{R}^n$ be a bijective linear map. What is the vector field $f_*v$? ...
2
votes
1answer
50 views

Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Show that {$x-f(x): |x|\leq 1$} is a compact set in $\mathbb{R}$.

Question: Let $f: \mathbb{R} \to \mathbb{R}$ be a continuous function. Show that {$x-f(x): |x|\leq 1$} is a compact set in $\mathbb{R}$. Attempt: If $f(x)$ is a continuous function, $f(x)$ is ...
2
votes
1answer
50 views

Please help make my infinite intersection proof more rigorous!

I am working on problems from my textbook for self-study and would like to know how I can make my proof more rigorous - I am having trouble expressing what I am thinking mathematically. I want to show ...
0
votes
3answers
87 views

Complete spaces

I was wondering whether there are other complete spaces than R. I understand that Q are not complete since there are irrational points missing, and one can construct a Cauchy sequence of rationals ...
2
votes
3answers
731 views

Is there a problem in studying analysis before calculus?

Is there a problem in studying analysis before calculus? Most people say that analysis is rigorous calculus, the university I'm studying teaches calculus first because they believe it's better for the ...
2
votes
2answers
246 views

Mean Value Theorem: Real Analysis

I need to show that $\dfrac{2}{\pi}<\dfrac{\sin(x)}{x}<1$ for $0<x<\dfrac{\pi}{2}$. I know I need to use the mean value theorem, would I just say that since $f$ is continuous in the ...
0
votes
1answer
47 views

topology simple bounded and limited point relation

If some set is bounded. there must exist an limited point? and if it is can you explain the detail? The bounded somehow related with the limited point? I''m trying to prove that every compact ...
1
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1answer
33 views

subset of a metric space and continuous functions

Let $$S = \{ x \in M \, | \, f(x) = a \}$$ be closed and $g$ a continuous function from $M$ to $\mathbb{R}$. If we change it to $f(x) \leq a$, then it won't be a line and S would not be closed ...
0
votes
1answer
30 views

Understanding Lyapunov boundaries and where are they used?

Reading some potential theory, my book almost always uses a rather strong regularity condition on the boundaries to be of class $C^2$. My book refers to a slightly weaker case of Lyapunov boundaries. ...
3
votes
2answers
103 views

Let $f(x)$ be a continuous function in $\mathbb{R}$, and let $(a_n)$ be a Cauchy sequence. Prove that $f(a_n)$ is a Cauchy sequence.

This is a question I've stumbled upon and the question asks me to prove something which I don't think is true. Question: Let $f(x)$ be a continuous function in $\mathbb{R}$, and let $(a_n)$ be a ...
1
vote
1answer
35 views

Under what conditions does integrating the normal vector along a boundary make no sense?

So suppose you have an open, simply-connected, and bounded subset $D$ of $\mathbb{R}^2$ with the boundary $\partial D$. I am interested in the integral of the normal vector along the boundary, i.e., ...
0
votes
1answer
40 views

metric space with one limit point

all sets are clopen in the metric space with a single limit point? I think it is not the case, but I'm not sure which space has a single limit point. I was thinking of convergent sequences and ...
2
votes
1answer
90 views

Definition of pull-back analogous to push-forward

Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Then the push-forward $f_*u$ is equal to $v$ ...
0
votes
1answer
54 views

Condition for $f$-related vector fields

Let $U\subseteq\mathbb{R}^n, V\subseteq\mathbb{R}^m$ be open subsets, and $f:U\rightarrow V$ a $C^1$ map. Let $u,v$ be vector fields on $U,V$ respectively. Show that $u,v$ are $f$-related if and ...
4
votes
1answer
573 views

Real Analysis and Statistics

What level of real analysis do you think is desirable for the study of statistics? I know that for many statisticians with applied focus, rigorous mathematics tend to give them a headache and I am ...
0
votes
1answer
223 views

Prove there are 3 real roots to this equation using Rolle's Theorem

I need to prove there are $3$ real solutions to $x^5 - 4x + 2 = 0$. I know $f(-2)$ is negative, $f(0)$ is positive, $f(1)$ is negative, $f(2)$ is positive so that by IVT there are at least $3$ roots. ...
4
votes
3answers
116 views

Conclusion about limit definition of e^a for a sequence of real numbers {a_n} converging to a?

I have seen this fact used in several demonstrations, but have never seen a proof of it. I believe the statement is: If $\{a_n\}$ is a sequence of real numbers such that $a_n \rightarrow a$ finite, ...
0
votes
1answer
57 views

Closed, Continuous, and Bounded

I need to prove or disprove this: Let $I$ be a closed interval. If $f\colon I\to \Bbb R$ is continuous on $I$, then $f$ is bounded on $I$.