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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-1
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0answers
24 views

How to calculate with complex arguments?

I've got problems with calculating the complex argument of a number. We defined it like this: Let $$f: \Bbb C \ -{0} \to \Bbb [0;2π)$$ for $x=0$ we define no argument. For $zw≠0$ it is: $$Arg(zw)= ...
-1
votes
0answers
14 views

Quick proof check

Intro to analysis by gaughan 1.4 #45. Show that if x is any real number, there is a sequence of rational numbers converging to x.
1
vote
3answers
28 views

How to show that this modification of Thomae's function is Riemann integrable

I am dealing with the function $f(x)=\begin{cases} \frac{1}{n} & \text{if }\frac{1}{n+1}<x<\frac{1}{n},\:n\in\mathbb{N},\\ 0 & \text{ otherwise.} \end{cases}$ I want to show it is ...
2
votes
1answer
20 views

Existence of a subset $S\subset\mathbb R$ s.t. $\forall a<b$, $S\cap [a,b]$ has Lebesgue measure $(b-a)/2$?

I am trying to either find an example of such a set, or prove that no such set exists. I know of examples of dense sets with measure $1/2$ on specific intervals, such as $[0,1]$, but I haven't been ...
2
votes
1answer
34 views

If $x$ is an isolated point of $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$.

Is the following proof valid? (Note: I know there is a post discussing this problem, but I am curious to see if my argument works). This problem is different from another post that is similar with ...
0
votes
2answers
21 views

Show that for any $x_0\in \mathbb{R}$, the one sided limits exist and that $f^+(x_0)\geq f^-(x_0)$.

Suppose $f(x)$ is a monotone increasing function defined for all $x\in \mathbb{R}$. Show that for any $x_0\in \mathbb{R}$, the one sided limits $$f^+(x_0)=\lim_{x\to x_0^+}f(x) \text{ and } ...
0
votes
1answer
37 views

Is every set in $S$ measurable with respect to the outer measure induced by $\mu$

Let $S$ be a collection of subsets of $X$ and $\mu : S \to [0, \infty]$ a set function. Is every set in $S$ measurable with respect to the outer measure induced by $\mu$ Here is how we defined outer ...
0
votes
2answers
38 views

Continuity of a function between metric spaces

I want to show: Let $(X,d)$ be a metric space and $A \subset X$ be a closed subset. Define $f: X \to \mathbb{R}$ by $$ f(x) = d(x,A) := \inf_{y\in A}d(x,y), \phantom{.} \forall x \in X.$$ Show ...
1
vote
1answer
21 views

$L^2$ convergence by the sequence of domain

Let $\Omega\subset \mathbb R^N$ be open bounded, smooth boundary. Assume $u\in L^\infty(\Omega)$. We know a sequence $u_n\in L^\infty(\Omega)$ such that $$ \sup_{n}\|u_n\|_{L^\infty}<+\infty $$ ...
5
votes
1answer
59 views

Are there sets of zero measure and full Hausdorff dimension?

I would like to ask the following: Are there "many" sets, say in the interval $[0,1]$, with zero Lebesgue measure but with Hausdorff dimension $1$? The motivation for this question is the ...
0
votes
3answers
30 views

Suppose {A} is a sequence that assumes only integer values, under what conditions does this sequence converge?

I don't know how to think of this question. Could it be a sequence that eventually repeats the same digit over and over again? How would I explain this?
0
votes
0answers
9 views

Proving a value-wise magnitude function is differentiable

Let $v: \mathbb{R^n} \rightarrow \mathbb{R^m}$ be a function such that $v(y) \neq 0,\forall y \in \mathbb{R^n}$ that is differentiable at $ x \in \mathbb{R^n}$. (a) Show that the value-wise ...
0
votes
1answer
43 views

Why is $U ⊂ \mathbb{R}^n$ open with respect to metric $d_p$ iff it is open with respect to metric $d_q$ for $q ∈ [1, ∞)$?

Let's say that for any $p ∈ [1, ∞)$ we have a distance function on $\mathbb{R}^n$ given by $$d_p(x, y) := \left(\sum^n_{j=1}|x_i - y_i|^p\right)^{\frac{1}{p}}$$ How would I show that a set $U ⊂ ...
1
vote
2answers
43 views

Infinite closed subset of $[0, 1]$ that does not have any subset of the form $[a, b]$ for $a< b$?

What is an infinite closed subset of $[0, 1]$ that does not have any subset of the form $[a, b]$ for $a< b$?
1
vote
2answers
19 views

Forming a sequence from a Cauchy Sequence

Let $(a_{n})$ be a Cauchy sequence. Is $c_{n} = (-1)^{n}a_{n}$ also a Cauchy sequence?
1
vote
2answers
33 views

Is There a Problem with This Professor's Proof Concerning Interior and Boundary Points?

Here is a professor's solution to the exercise which states, " Prove that if $x$ is an isolated point of a set $S \subseteq \mathbb{R}$, then $x$ is a boundary point of $S$." The professor derived ...
0
votes
1answer
22 views

Non-monotonically decreasing flow whose limit is $\vec{0}$

I'm trying to come up with $x'=Ax$, which is a system of linear differential equations, whose flow satisfies $\lim\limits_{t\to\infty} \lvert e^{tA}x\lvert = 0$ for all $x\in \mathbb{R}^n$, but ...
5
votes
1answer
61 views

If $\sum a_n$and$\sum b_n$diverge, can$\sum \min\{a_n,b_n\}$converge? [duplicate]

Do there exist sequences $\{a_n\}$ and $\{b_n\}$ satisfying all of the following properties? $a_n>0$ and $b_n>0$ $\{a_n\}$ and $\{b_n\}$ are both decreasing $\sum a_n$ and $\sum b_n$ both ...
1
vote
0answers
25 views

inverse function theorem problem

I am interested in finding a formula for the inverse of the function $$f(x,y) = (x^2 + y^2, xy)$$ which works on the set $S = \{(x,y) \in \mathbb{R}^2 : -x < y < x\}$. I determined that the ...
0
votes
1answer
44 views

Bounding a strange function

Let $a>0$, show that for $x>0$, $1<f(x)<2$, where $$f(x)=\frac{1}{\sqrt{1+x}}+\frac{1}{\sqrt{1+a}}+\sqrt{\frac{ax}{ax+8}}$$ I could take the derivative, find the maximum of the function ...
0
votes
1answer
12 views

Representing $C(X)$ as multiplication operators on $L^p$

Suppose that $X$ is a compact Hausdorff space and I represent $C(X)$ isometrically in $B(L^p(X,\mu))$ as multiplication operators for some finite positive regular Borel measure $\mu$. If I remember ...
0
votes
0answers
27 views

Let $S$ be the set . Which of following are true?

Let $S =\{\frac{1}{3^m}+\frac{1}{7^n}$ , where $m,n \in \mathbb N\}$ Then A.$S$ is closed B.$S$ is not open C.$S$ is connected D.$0$ is a limit point of S I see that $0$ is limit point of $S$ but ...
1
vote
2answers
33 views

Find an open set whose preimage is not open for $f$ [on hold]

Consider $f(0)=0, f(x) = \cos\left(\frac{1}{x}\right)$ otherwise in $\mathbb{R}$ Find an open set whose preimage is not open. I think I need an open set not including $0$.
3
votes
3answers
42 views

A footnote about outer measure

This is the theorem about in Royden's real analysis book. And in the book there is a footnote I am confusing: Can anyone help me understanding it with examples~~~
1
vote
0answers
74 views

Another way of doing integration

What's your option for calculating this integral? No full solution is necessary, it's optional as usual. Calculate $$\int_0^1 \frac{2 \zeta (3)\log ^3(1-x) \text{Li}_2(1-x) }{x}-\frac{2 \zeta (3) ...
1
vote
1answer
35 views

Weak convergence and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ imply norm convergence.

Consider a $\sigma$-finite measure space $(X,A,\mu)$ and $f,f_n\in L^p(\mu)$ with $1<p<\infty$. If $f_n \stackrel{w}{\to} f$ and $\lim_{n\to \infty} \|f_n\|_{L^p}=\|f\|_{L^p}$ hold, then ...
4
votes
3answers
115 views

How do I show that if $f$ is bounded and integrable on $\mathbb{R}$, then $g(t) = \int_t^{t+1} f(x) dx$ is continuous?

Usually functions of the form of $g(t)$ tell me I should use the fundamental theorem of calculus, but I don't think that applies here because I'm not given that $f$ is continuous. I know from the ...
0
votes
1answer
34 views

Prove, that for any $0<a<1$, $-\frac{a}{1+a}<\ln(1-a)<-a.$

Prove, that for any $a>0$, $$a>\ln(1-a)>\frac{a}{1+a}.$$ Prove, that for any $0<a<1$, $$-\frac{a}{1+a}<\ln(1-a)<-a.$$ Proof of 1: We will prove (1) by doing a proof by ...
0
votes
0answers
13 views

Is this part of a known sequence?

while trying to express as an infinite sum the function $t^x/\Gamma(x)$ I came across some coefficients of the form $a_0=1$ $a_1=-\psi^{(0)}(1)$ $a_2=[\psi^{(0)}(1)]^2-\psi^{(1)}(1)$ ...
4
votes
0answers
43 views

Lebesgue versus Riemann integrable

Can a Lebesgue measurable function be modified on a set of first category so as become continuous except on a set of Lebesgue measure zero? OR Can a Baire-measurable function be modified on a set of ...
2
votes
1answer
69 views

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

$$\lim_{x \to \infty} \frac{(\frac x n)^x e^{-x}}{(x-2)!}$$ where $x$ is $\mathbb N$-valued and $n$ is some nonzero real number. Wolfram seems to give $0$ for different values of $n$ that I tried. ...
3
votes
0answers
22 views

Convergence of average of translates of a function

Short version for people who don't like reading: Let $f\colon\mathbb{R}\to\mathbb{R}$ be $1$-periodic and $L^1$ on one period (or perhaps: measurable and bounded). Is it true that, for almost ...
1
vote
1answer
64 views

If $f$ is a Riemann integrable function on $[a,b]$, is there always a Riemann sum whose value is greater than or equal to the value of the integral?

Can you give me a hint as to how to show this? I need it for a homework problem I'm working on, and I'm technically not supposed to know about the Upper Riemann sum of a function, although I've ...
0
votes
1answer
21 views

A faithful positive Radon measure

Let $X$ be a locally comapct and Hausdorff space. We say a positive Radon Measure on $X$ is faithful if $$0\leq f ~~~,~~~\int fd\mu=0\rightarrow f(x)=0 ~~\forall x\in X$$ Q: True or false: If there ...
7
votes
2answers
121 views

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$

Let $f:\mathbb{R}\rightarrow\mathbb{R}$ be a function such that $f'(x)$ is continuous and $|f'(x)|\le|f(x)|$ for all $x\in\mathbb{R}$. If $f(0)=0$, find the maximum value of $f(5)$. $f'(x)=f(x)$ ...
2
votes
2answers
92 views

About $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$

Suppose $M$ is map from vector space $X$ to vector space $Y$, $M(0) =0$, and $M(\frac{x+y}{2}) = \frac{1}{2}(M(x) + M(y))$. Does this mean that $M$ is a linear map? If not, could someone please give ...
0
votes
2answers
15 views

Find PDF on $[0,6]$ such that $P([1,3]) = 0.5$

Find a probability density function $f$ on $[0,6] \subset \mathbb{R}$, such that $\mathbb{P}([1,3]) = 0.5$ That is we need to find an $f$, such that $\int_{[0,6]} f(x)dx = 1$ and $\int_{1}^{3} ...
-2
votes
1answer
28 views

Change Integration order [on hold]

Let $f: [0,b] \times[0,b] \longrightarrow \mathbf {R}$ be continuous. Show that \begin{equation} \int_0^b\int_0^xf(x,y)~dy~dx = \int_0^b\int_y^bf(x,y)~dx~dy \end{equation}
11
votes
0answers
91 views

References about Iterating integration, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$

Are there any references that discuss Iterating integration in general, $\int_{a_0}^{\int_{a_1}^\vdots I_1dx}I_0\,dx$, conditions in which they converge, some special values, some special tricks to ...
0
votes
0answers
18 views

Local maximum of $(2^{xy}{z \choose y})^{z+1}$

I have an optimization problem where I need to calculate the maximum of the following function $$ f(x,y,z) = (2^{xy}{z \choose y})^{z+1} $$ where $$ (z+1)(a+y(\lceil{\log_2{(z+1)}}\rceil+x))\leq C $$ ...
2
votes
0answers
34 views

Reciprocal of a limit that goes to infinity

Lets say we have a limit $\lim_{n \rightarrow \infty} \frac{a_n}{b_n} = +\infty$, then is it safe to assume that $\lim_{n \rightarrow \infty} \frac{b_n}{a_n} = 0$?
1
vote
0answers
16 views

Integral of sums of basic trigonometric polynomials

My textbook (RCA, Rudin) asserts (in the proof of theorem 5.15) that $$ \lim_{n\to\infty}\lVert{\sum_{k=-n}^n}e^{ikt}\rVert_1=\infty. $$ Why is this true? I tried using Euler's formula to reduce the ...
0
votes
1answer
24 views

Approximating a Riemann integrable function using a continuous function

Let $f$ be Riemann integrable on $[a,b]$. Show that for every $ε > 0$, there is a continuous function $g$ on $[a,b]$ such that $$\int_a^b |f(x)−g(x)|\mathrm dx < ε. $$
4
votes
5answers
261 views

How to find a differentiable function with bounded derivative satisfying some boundary conditions?

I am trying to find an example, preferably an explicit one, of a differentiable function $g:\mathbb{R}\rightarrow \mathbb{R}$ satisfying the following conditions: $\displaystyle g(0)=0, g(1)=1, ...
1
vote
0answers
16 views

A Possible Logical Problem with Showing that $x$ is a Boundary Point Whenever It is an Isolated Point

Prove: If $x$ is an isolated point of a set $S$, then $x \in \mathrm{bd} \, S$. I have two ways to solving this problem, but I believe the first one has a logical issue which I will explain below. ...
19
votes
1answer
212 views
+50

Show that $ \lim\limits_{n\to\infty}\frac{1}{n}\sum\limits_{k=0}^{n-1}e^{ik^2}=0$

TL;DR : The question is how do I show that $\displaystyle \lim_{n\to\infty}\frac{1}{n}\sum_{k=0}^{n-1}e^{ik^2}=0$ ? More generaly the question would be : given an increasing sequence of integers ...
1
vote
2answers
26 views

Could fast or irregular oscillations make Lebesgue integral fail?

Let's consider real measurable functions defined in a bounded interval. As long as a function is bounded, oscillations at least cannot make the volume under the graph of the function infinite. But I'm ...
0
votes
2answers
41 views

Prove that $\frac{(a+1)^k}{a^k+1} \leq 2^{k-1}$

if $ k \geq 0$ Prove that $\frac{(a+1)^k}{a^k+1} \leq 2^{k-1}$ Proof (induction) let k = 1 then $$\frac{(a+1)^1}{a^1+1} \leq 2^{1-1}$$ , implies that $$ 1 \leq 1$$ so p(1) is true. Inductive step ...
2
votes
1answer
62 views

Prove that the following statements are equivalent characterizations of continuity

Let $f: (X,d) \rightarrow (Y, d')$ be a function. Prove that the following are equivalent: $f$ is continuous . For every $A \subset X$, $f(cl(A)) \subset cl(f(A))$. For every closed set $B$ in ...
6
votes
1answer
90 views

the elements of Cantor's discontinuum

Let $(A_n)_{n \in \mathbb{N}}$ the sequence of subsets of $\mathbb{R}$, given by $A_0 := \bigcup_{k \in \mathbb{Z}}[2k, 2k + 1]$ und $A_n := \frac{1}{3}A_{n-1}$ for $n ≥ 1$. Also, we define $$ A := ...