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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15 views

Cauchy Sequences, not converging to zero

True or False? If $\{x_n\}$ and $\{y_n\}$ are Cauchy and $x_n + y_n > 0$, for all $n\in\mathbb{N}$, then $\left\{\frac{1}{(x_n + y_n)}\right\}$ cannot converge to zero. I believe the claim to be ...
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1answer
31 views

alternating series $\sum(-1)^na_n$ is divergent, then, is $\sum A_k$ divergent?

An alternating series $\sum\limits_{n=1}^\infty (-1)^na_n$ is divergent , $a_n\geq0$, and $\lim\limits_{n\to\infty}a_n=0$. Could we conclude that $\sum\limits_{k=1}^\infty A_k$ is divergent, too ? ...
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7 views

Measuring Unsigned Simple Functions

I was hoping that someone would be able to help me solve this problem regarding simple functions and their measure. Show that an unsigned function $f: \mathbb{R}^d \to [0, +\infty]$ is a simple ...
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1answer
26 views

Connected Sets on Metric Spaces

I'm taking a first course in real analysis, and we're using Rudin's Principles of Mathematical Analysis as our main (only) book. In chapter two, Rudin discusses basic topology from the point of view ...
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10 views

I need some clarification on the term of “measurable on” and “continuous on”

I run across theorems similar to this one: "If $f$ is a complex measurable function on $X$, there is a complex measurable function on $X$ called $\alpha$ such that $\vert\alpha\vert=1$ and ...
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1answer
24 views

Distance between a point and a closed set in metric space

Here is what I am thinking. Let (X,d) be a metric space and let C be a closed subset of X. Fix any poin p in X. Then, there exists a point q in C such that d(p,q) = distance(p,C). I think this ...
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16 views

Dependent Expectation in Random Numbers Illustrated by Prime Repetition in Pi

When approximating Pi, appending each numerical digit as you refine, what is the first repetition of a four-digit prime number? For instance the first repetition of any one-digit number in the ...
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2answers
23 views

How to approximate a globally Lipschitz function by differentiable functions with bounded derivatives?

for some positive integer $d \geq 1$ I have a globally Lipschitz continuous function $f \colon \mathbb{R}^d \to \mathbb{R}$ with Lipschitz constant $1$ and would like to approximate it by a sequence ...
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0answers
27 views

Inequality in $\mathbb{Z}^2$

Let $k=(k_1,k_2)\in\mathbb{Z}^2$. Denote $|k|\leqslant n$ when $|k_1|,|k_2|\leqslant n$. I need help to show $$|\sum_{k+l+m=0}_{|k|,|l|,|m|\leqslant ...
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1answer
30 views

Continuous iff Oscillation is zero

For a bounded function $f:D\subset \Bbb R^n \rightarrow \Bbb R$, $b$ in $\Bbb R^n$, and a real number $\delta>0$. Define the following: $M(f,b,\delta)$=sup{f(x)$: x$ in $D$ and ...
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30 views

P-norm Unit Ball

Proof that for $0<p<1$, $p\in \Bbb{R}$ $$\|(x,y)\|_p=(|x|^p+|y|^p)^{\frac{1}{p}}$$ doesn't define a norm in $\Bbb{R}^2$. However, $$d_p((x_1,x_2),(y_1,y_2))=\sum_{i=1}^2|x_i-y_i|^p$$ defines a ...
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0answers
10 views

Why all closed intervals of $R$ is a semi-algebra?

How the class of all closed intervals can be a semi-algebra?
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20 views

How to prove that a piecewise defined function is Borel measurable?

Given $$ϕ_n(t)=\begin{cases}k_n(t)δ_n\quad &\text{ if $0≤t< n$} \\ n \quad &\text{ if $n≤t $}\end{cases}$$ show that $ϕ^{-1}_n((a,∞])$ is an interval for any real $a$, hence $ϕ_n(t)$ ...
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28 views
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60 views

Prove that $X$ is complete but not inner product, and vice versa

Let $X$ be the space $C[0, 1]$ under the norm $||·||_{p}$ for $1 \leq p \leq \infty$. (a) Show that $X$ is complete for $p = \infty$, but it is then not an inner product space. (b) Show that $X$ is ...
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0answers
37 views

Is there problem follow my definitions to define $\mathbb{R}$ [on hold]

There are definitions like this: A. operates $-$ ,$+$, $*$, / $\vec{d}(x,t)=t-x$ B. items $\mathbb{Z}=$ -j, ... ,-i ,... 1, 2, ... , i... j..., $t$ 's neighborhood $N(t)$ is ...
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1answer
18 views

Prove that for $0<p<1$, $|x-y|^p$ is a metric space on $R^{n}$

Define the function $f_p : R^{n} → R^{n}$ for $n ≥ 2$ by $f_p(x) = \sum_{k=1}^{n} |x|^{p}$. Show that for $0<p<1$, we get $d_b(x,y) = f_p(x-y)$ is a metric on $R^{n}$. I tried to use ...
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0answers
13 views

Mean value theorem and Multi-valued functions

Let a point $x$ map to a set of points $\{y | y \in U \}$ where $U \subseteq \mathbb{R}$ or $U \subseteq \mathbb{C}$. Can MVT be generalized to multi-valued functions ?
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1answer
30 views

Is there exist an additive but unbounded function?

I just learned that the function that is additive and bounded near $0$ on Real has the only form of $f(x)=cx$, where $c$ is a constant number. We say that a function $f$ is additive iff ...
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0answers
14 views

A Difficult combinatorial optimization problem

Let $\mathcal{J}$ be a closed, bounded, compact, convex set in $\mathbb{R}^L$. (Notations: vector $\mathbf{x}$ is denoted in bold letters and its $i^{th}$ co-ordinate is denoted as $x_i$. ...
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1answer
43 views

Techniques to solve such a PDE

I have the eigenvalues problem on $[0,\pi] \times [0,2\pi]$ $$\left(\frac{1}{\sin\theta}\frac{\partial}{\partial \theta} \left[\sin\theta \frac{\partial}{\partial \theta}\right] + ...
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43 views

Show that there exists $\xi\in [a,b]: f(\xi)=\xi$.

Let $a,b\in\mathbb{R},~a<b$ and consider $f\colon[a,b]\to [a,b]$ continuous. Show that $f$ has a fixed point. i.e. that there exists a $\xi\in [a,b]$ with $f(\xi)=\xi$. My idea is to ...
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28 views

Could some one please explain how to make sense of this theorem intuitively? [on hold]

A number u is the supremum of a nonempty subset S of R if and only if u satisfies the conditions: (1) s u for all s in S, (2) if v
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1answer
35 views

Proving Archimedes Sequences Equal $\pi$.

I encountered the following problem in my text An Introduction to Analysis by William Wade. It was the last problem in the section and has an * next to it. I'm not sure if this indicates a challenge ...
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1answer
28 views

Condition for derivative sequence to converge?

Let $E_n(R)$ be a sequence of function that converge to $E(R)$, When we can said that $\dot{E}_n(R)$ converge to $\dot{E}(R)$. I can assume: uniform converge of $E_n$ to $E$. $E(R)$ convex. ...
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3answers
53 views

Prove this function is pointwise continuous

Prove that the function $f\colon(0,1)\cup(1,2)\mapsto\mathbb{R}$ is continuous at all points in its domain. $$f(x)= \begin{cases} x : 0<x<1\\ 0 : 1<x<2 \end{cases}$$ The graph of $f$:
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2answers
31 views

Proving that a set $A$ is dense in $M$ iff $A^c$ has empty interior

Prove that a set $A$ is dense in a metric space $(M,d)$ iff $A^c$ has empty interior. Attempt: I think I proved the converse correctly, but I'm not sure how to start the forward direction. ...
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1answer
23 views

Why $f (x):= \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)}$ only belongs to $L^2(0, \infty)$

This is a result given in Royden and Fitzpatrick (p. 143). Show that $$ \int_0^\infty \left[ \frac{1}{\sqrt{x}\left(1+\left|\ln x\right|\right)} \right]^p < \infty $$ if and only if $p=2$. That ...
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1answer
33 views

Using proof by contradiction to prove 'Why are discontinuities of monotonic f:(a,b)→R countable?' [on hold]

I know it is a theory 4.30 in Walter Rudin's textbook, and there is problems like How to show that a set of discontinuous points of an increasing function is at most countable . This is my ...
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2answers
39 views

An intuitive idea about the limit of a continuous function: Is it correct?

Let's assume that we have a function $f(x)$ whose limit at $c$ is given as $\lim_{x \to c}f(x) = f(c)$, such that it is continuous at $c$. For this limit, we have left and right side limits $\lim_{x ...
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1answer
26 views

How to know if a space has a convergent subsequence?

So this is something I have been struggling with lately... how do we generally know that a space/set has a subsequence that converges? The current one I am struggling with is the space of sequences ...
2
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1answer
28 views

limit of greatest integer less than

Does $\lim_{x \to \infty} [1- \frac{1}{x}] = 0$ ? Where $[x]$ is the greatest integer less than or equal to $x$ for $x\in \mathbb{R}$.
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1answer
22 views

What are the epis in Met?

I have an assignment to precisely describe epimorphisms and monomorphisms in Met (category whose objects are Metric spaces and whose morphisms are contractions). I have shown that Mono $\iff$ ...
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1answer
34 views

Proving a sequence is increasing and converging as $n\to \infty$.

Suppose that $x_0 \in (-1,0)$ and $x_n=\sqrt{x_{n-1}+1}-1$ for $n \in \mathbb N$. Prove that $x_n \uparrow 0$ as $n\to \infty$. What happens when $x_0 \in [-1,0]$? Before this, the problems I did had ...
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3answers
23 views

Prove that $d_n$ is a Cauchy sequence in $\mathbb{R}$

Let $(x_n$) and $(y_n)$ be Cauchy sequences in $\mathbb{R}^n$ , i.e. lim$_{n,m}$ |$x_n$ − $x_m$| = $0$ and lim$_{n,m}$ |$y_n$ − $y_m$| = $0$. For each n, let $d_n = |x_n − y_n|$. Prove that $d_n$ is a ...
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1answer
13 views

Prove that $F'(x) = \sum_{n=1}^\infty F_n'(x)$ almost everywhere.

Suppose $F_n$ is a sequence of increasing non-negative right continuous functions on $[0,1]$ such that $\sup_n F_n(1) < \infty$. Let $F = \sum_{n=1}^\infty F_n$ and suppose that $F(1) < \infty$. ...
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44 views

How to prove that $b^{x+y} = b^x b^y$ using this approach?

Fix $b>1$. If $m$, $n$, $p$, $q$ are integers, $n > 0$, $q > 0$, $r = m/n = p/q$, then I can prove that $(b^m)^{1/n} = (b^p)^{1/q}$. Hence it makes sense to define $b^r = (b^m)^{1/n}$. I ...
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1answer
34 views

A bit stuck on proving a function, $ f : [-1,1] \rightarrow \mathbb{R} $ is differentiable from the definition.

I have the following function $ f_n : [-1,1] \rightarrow \mathbb{R} $, $$ f_n (x) = |x|^{1+\frac{1}{n}} $$ I need to prove its differentiability from the definition $$ \lim_{x \rightarrow c} \frac{ ...
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51 views

How to find if an integral is possible to compute: Failing to solve integral for quadratic functional

I am trying to solve the below integral, and no computational method seems to be capable of solving this, nor can I do it by hand. Any ideas? $$\int_{t_0}^{t_1}[a(t)((2\dot{x^*}\dot{\eta} + ...
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Fourier Transforms of $L^1$ functions

Suppose that $f_n$ and $f$ are $L^1(\mathbb R^n)$ functions with $f_n \to f$ in $L^1$ sense. Then is it true that their Fourier transforms defined as $$ \hat f(\xi) := \int_{\mathbb R^n} ...
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12 views

Eliminating variables in convex program

This is a basic convex optimisation question. I have the following problem: $$\max_{\substack{t\le e\\ At\le b}} e^\top t$$ How do I find the optimum $t^*$? I write the KKT conditions, get ...
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1answer
18 views

Is the split normal distribution analytic on $\mathbb{C}$?

I wonder if the split normal distribution which expressed as following is analytic on $\mathbb{C}$ or not? $ p(x)= \left\{ \begin{array}{l l} \frac{2}{1+\gamma} \cdot \frac{1}{\sqrt{2 \pi}} ...
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2answers
82 views

Closed form of $\sum_{n=1}^{\infty} \frac{J_0(2n)}{n^2}$

I'm new in the area of the series involving Bessel function of the first kind. What are the usual tools you would recommend me for computing such a series? Thanks. $$\sum_{n=1}^{\infty} ...
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3answers
100 views

Limit of the sequence $\frac{1^k+2^k+…+n^k}{n^{k+1}}$ [duplicate]

How would someone find the limit of the sequence $a_n = \frac{1^k+2^k+...+n^k}{n^{k+1}}, k \in \mathbb{N}$ as $n$ goes to Infinity? Can someone give me maybe a hint where to start?
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1answer
19 views

Equivalence of convergence in products of sequences.

Let $\{x_{n}\}_{n \in \mathbb{N}}$ and $\{y_{n}\}_{n \in \mathbb{N}}$ be two convergent sequences. I am trying to show that \begin{align} \lim_{n \rightarrow \infty} \frac{1}{n+1}\sum_{k = 0}^{n} ...
6
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2answers
97 views

Computing in closed form $\sum_{n=1}^{\infty}\frac{\operatorname{Ci}\left(\frac{3}{4}\zeta(2) \space n\right)}{n^2}$

What tools would you recommend me for computing the series below? $$\sum_{n=1}^{\infty}\frac{\operatorname{\displaystyle Ci\left(\frac{3}{4}\zeta(2) \space n\right)}}{n^2}$$ I lack the starting ...
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3answers
68 views

Is infinity well defined?

Let us consider two functions $f\left(t\right) = \frac{1}{t^2}$ and $g\left(t\right) = t^2$. We want to find the value of these two functions as $t\rightarrow \infty$. For the function $f$, this value ...
1
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1answer
15 views

If $f$ is $C^1(U)$) , are $D_i f_j$ where $i=1,\ldots,n$ and $j=1,\ldots,m$ are all continuous on $U$?

$f$ is a function from an open set $U$ in $R^n$ to $R^m$ then $f=(f_1,f_2,\ldots,f_m)$, I am confused whether the following are true: If $f$ is continuous on $U$, does that imply that ...
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0answers
39 views

Prove $f$ is Riemann integrable

Let $\{b_n\}$ be decreasing sequence converging to $0$. Define $f$ in terms of $b_n$ as follows $f(0)=1$ and $f(x)=0$ if $x$ is irrational, and $f(\frac{m}{n})=b_n$ if $x=\frac{m}{n}$ with ...
0
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1answer
49 views

I dont understand how this Maclaurin series got manipulated into looking like this other Maclaurin series. Help Please

I have been reading a book on approximating $e$ and there is a couple lines that I am stuck on. Here they are: $x$ln$(1+\displaystyle\frac{1}{x}) = 1 - \displaystyle\frac{1}{2x} + ...