Tagged Questions

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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3
votes
1answer
28 views

Evaluate the limit using only the following results

Let $u_2>u_1>0$ and also let $u_{n+1}=\sqrt{u_n u_{n-1}}$ for all $n \geq 2$. Then prove that the sequence $\{u_n\}$ converges. For this, use of only the following results is permissible, ...
0
votes
1answer
19 views

Proving a point exists on a twice differentiable function.

Problem: Suppose that $f$ is twice differentiable on $(a,b)$ and that there are points $x_1\lt x_2\lt x_3$ in $(a,b)$ such that $f(x_1)\gt f(x_2)$ and $f(x_3)\gt f(x_2)$. Prove that there is a point ...
0
votes
1answer
14 views

Show that zero sequences satisfy the following equation

I am working on the following problem and got puzzled: $\\$ Show that every zero sequence $(a_n), a_n \neq 0$ satisfies the equation: $$ \lim_{n \rightarrow \infty} \frac{\sqrt {1 + a_n} - 1}{a_n} = ...
1
vote
1answer
31 views

Finding all continuous and discontinuous points of composite functions

Let $f(x) = \operatorname{sgn}(x)$ and $g(x) = 1 + x^2$. How do I go about finding all the continuous and discontinuous points of the functions $f\circ g$ and $g\circ f$ ?
2
votes
2answers
53 views

Finding the $nth$ partial sum for $e^{-n}$

Here is the question: $$\displaystyle \sum_{n=1}^{\infty} e^{-n}$$ Instead of using the formula of $\large\frac{1}{1-r}$ I want to try to get the partial sums. $S_1 = e^{-1}$ $S_2 = e^{-1} + ...
6
votes
2answers
30 views

Deriving the analytical properties of the logarithm from an algebraic definition.

Definition: The base $a$ logarithm ($a\in]0,1[\cup]1,+\infty[$) is the continuous function defined by: $\log_a(xy)=\log_a(x)+\log_a(y)~~\forall x,y>0$ and $\log_a(a)=1$ If I used this definition ...
0
votes
1answer
13 views

Derivative of Lebesgue integral function at the endpoints

Let $f$ be a non-decreasing Lebesgue-integrable real function on $[a,b]$. I read in Kolmogorov-Fomin's Элементы теории функций и функционального анализа (p. 340 here) that $$\lim_{h\to ...
0
votes
3answers
63 views

Prove that the function is continuous at n where n is an integer, but discontinuous elsewhere.

I'm working on my self study again, and I'm given a function $f(x)=\sin\pi x$ , where $x$ is rational and $f(x) = 0$ when $x$ is irrational. How do I prove that the function is continuous at $n$, ...
2
votes
1answer
19 views

Formal explanation for this change of integration [duplicate]

Formally, why is true that $$\int_0^\infty \int_{x}^\infty f(x,y)dy dx =\int_0^\infty \int_{0}^y f(x,y)dx dy $$ ? I know and understand perfectly the geometric interpretation, and with that, I´m well ...
0
votes
1answer
49 views

If $f$ is continuous, then $\lim\limits_{n \rightarrow \infty} \int^b_a n(f(x+ 1/n)-f(x)) \lambda(dx) = f(b)-f(a)$

Consider a continuous function $f: \mathbb R \rightarrow \mathbb R$ and define $f_n: \mathbb R \rightarrow \mathbb R$ by $f_n(x) = n(f(x+1/n)-f(x))$. I want to show that for $a < b \in \mathbb R$ ...
0
votes
2answers
24 views

Divergent $\epsilon - N$ Proof

Let: $\frac{n+1}{\sqrt{n}}$ I'm having trouble proving this with an $\epsilon - N$ proof, I know that it's divergent but all of the divergence sequences that I have come across are bounded and just ...
5
votes
3answers
58 views

Divergence of $a_{n+1}=\sqrt{2a_n+3}$?

I am wondering what I am missing from my proof. I would like to show that the limit of the sequence $$a_{n+1}=\sqrt{2a_n+3},\,\, a_1=4,$$ goes to $\infty$, as $n \rightarrow \infty$. Is there ...
-4
votes
0answers
486 views

Help regarding $p=np$ [on hold]

Isn't it just true whenever $n=1$ and/or $p=0$? Or am I missing something here? Do we have any bounds on $p$ or $n$ or even $n\cdot p$ I am just learning about some computer science so please be ...
2
votes
1answer
42 views

Periodicity over the prime indices of a multiplicative sequence implies periodicity?

I have a real sequence $(a_p)$ indexed by the prime numbers which takes values -1, 0, or 1, having the property that $a_p=a_q$ whenever $p\equiv q \pmod m$, where $m$ is a fixed integer $>2$. I'm ...
0
votes
1answer
19 views

Riemann Integrables

Let $f: [a,b] \rightarrow \mathbb{R}$ and $g: [a,b] \rightarrow \mathbb{R}$ Prove the following. $f \le M \implies \int^b_a f(x)dx \le M(b-a)$ $f+g \in \mathbb{R[a,b]}$ and $\int^b_a [f(x)+g(x)]dx ...
1
vote
2answers
39 views

For every real number $a$ and for every $\epsilon > 0 $ there are infinitely many rationals between $a$ and $a$ + $\epsilon $

What i'm trying to show is that for every real number $a$, in his $\epsilon$ surrounding i can find infinitely many rationals. If i define set $A=\{ x_0|a<x_0<a+\epsilon\}\bigcup ( \bigcup_{n ...
1
vote
6answers
56 views

Examples of dense sets in the complex plane

We know that the set $\left\{a+ib:a,b \in \mathbb{Q} \right\}$ is dense in $\mathbb{C}$. Could one give other examples of dense sets in the complex plane?
1
vote
1answer
37 views

A sequence of truncates of $f$

If $f$ is measurable and $A>0$ then the truncation $f_{A}$ defined by: $$f_{A}(x)=\begin{cases} f(x)&\text{if $\left | f(x) \right |\leq A$}\\ A&\text{if $ f(x)> A ...
-2
votes
4answers
102 views

How to prove $x^2+x$ is not uniformly continuous on $\mathbb{R}$? [on hold]

I think it is continuous? How to prove $x^2+x$ is not uniformly continuous on $\mathbb{R}$?
0
votes
1answer
32 views

Integral of the log is less than the integral of the log of the average value

This is an interesting property that I came across while reading an old proof on this website. The poster didn't really explain it, so I thought I might ask. We suppose $u$ is a positive measure on ...
3
votes
1answer
36 views

Compactness and uniform equicontinuous family

Suppose $\mathcal{F} \subset C(A)$ be a family of continuous functions with domain $A$. If $\mathcal{F}$ is pointwise equicontinuous, is it true that $\cal F$ is uniformly equicontinuous? I ...
4
votes
3answers
36 views

Can $\varepsilon$-$\delta$ definitions be used to find a limit or only to verify?

so I was wondering if there is any part of the $\varepsilon$-$\delta$ definition of the limit that offers any insight on how to find the limit of a function, or if this is something you are supposed ...
1
vote
2answers
38 views

Delta-Epsilon Proof of Continuity of a Function

Define $f\colon \mathbb{R} \times \mathbb{R}\to\mathbb{R}$ as $\dfrac{xy}{x^2 + y^2}$ for $(x, y) \neq (0, 0)$ and set $f(0, 0) = 0$. Determine whether $f$ is continuous. Please keep in mind that I'm ...
0
votes
1answer
24 views

Sequential criterion for functional limits proof in the opposite direction

Let $f: A\to\mathbb{R}$ Given $c$ is a cluster point in $A$. Prove that the following statements are equivalent: (a) The function $f$ does not have a limit at $c$. (b) There exists a sequence ...
1
vote
2answers
45 views

Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$.

Let $S$ be a bounded set. Prove there is an increasing sequence ($s_n$) of points in $S$ such that $\lim s_n = \sup S$. Note: If $\sup S$ is in $S$, it’s sufficient to define $s_n = \sup S$ for all ...
1
vote
1answer
20 views

If a function is continuous on $\mathbb R,$ does it follow that it is uniformly continuous on $(-1,1)?$

I've been trying to think of counterexamples but the ideas I've had so far like $1/(x+1)$ and $\sin(1/(x+1))$ don't work because those aren't continuous on all of $\mathbb R.$
0
votes
0answers
12 views

If $y$ is a limit point of a countable union of sets then is $y$ a limit point of one of the sets in the union?

Background I am attempting to construct a compact set whose limit points form a countable (not finite) set. The set I have constructed is as follows: Let $K={0} \cup \{\frac{1}{n} \}_{n=1}^{\infty}$ ...
2
votes
1answer
39 views

Nowhere differentiability of Weierstrass function

It's again from Tao's book. Let $f:\mathbb{R}\rightarrow \mathbb{R}$ be the function $$f:= \sum_{n=1}^\infty 4^{-n} \sin(8^n\pi x)$$ Show that for every 8-dyadic interval ...
3
votes
1answer
30 views

Proving limits with existing results

So in my lecture yesterday I learnt how to prove that $\lim_{x\to +\infty} \left(1 + \frac{1}{x}\right)^{x} = e$, but I'm lost as to how to apply the results to prove limits. Any help would be greatly ...
0
votes
0answers
10 views

Subadditive sequence over n and convergence. [on hold]

For a subadditive sequence (a), how do I show that (a/n) converges?
0
votes
1answer
20 views

How to prove continuity of addition over weird metric? Edit: Ignore this. Errors in the problem definition.

Let $f: R \times R \rightarrow R$ and let the metric over $R$ be $d(x,y)=|x-y|$ and let the metric in $R \times R$ be $d_2((x,y),(a,b))= ((x-y)^2+(a-b)^2)^{1/2}$. I believe I understand how to ...
2
votes
1answer
28 views

Help formalizing this proof about a continuous, one-one function.

I'm having a bit of trouble getting the language on this proof right, though I think I have the idea correct. I have the function $f\colon D \rightarrow {\bf R}$ where $D = [a,b]$. The function is ...
1
vote
1answer
17 views

Monotonically decreasing sequences

Suppose $(a_n),(b_n)$ be positive sequences such that $(a_n)$ decreases to $0$, monotonically. If lim$_{n\rightarrow \infty}\frac{b_n}{a_n}=1$, does it imply that $(b_n)$ decreases to $0$ ...
2
votes
1answer
49 views

Metric spaces in topology

$(X,d)$ be a metric space. Given $\epsilon >0$ there is a non empty finite subset $X_\epsilon\subset X$ such that for every $x\in X$, we have inf$\{d(x,p):p\in X_\epsilon\}\leq\epsilon$. 1) Show ...
0
votes
1answer
14 views

Prove the function $u(x):=1-|x|^{2-N}$ is in $W^{2,p}$ on $\{x\in \mathbb R^N;\,\,|x|>1\}$

This is exercise 10.11 from Leoni's book. Take $\Omega:=\{x\in \mathbb R^N;\,\,|x|>1\}$ and let $$u(x):=1-|x|^{2-N}$$ for $N\geq 3$. I am trying to prove that $\frac{\partial^2u}{\partial ...
2
votes
2answers
59 views

$\lim_{n \rightarrow \infty} \frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} \right)^n} = e^{-\frac{x^2}{6}} $

I am wondering about a limit that wolframalpha got me and that you can find here wolframalpha It says that $$\lim_{n \rightarrow \infty} \frac{\sin^{n}(\frac{x}{\sqrt{n}})}{\left(\frac{x}{\sqrt{n}} ...
1
vote
1answer
30 views

The dilogarithm function. Question on an identity of it

Upon reading a journal article about manipulating series using the dilogarithm function, I have a few questions. But before I ask them, let me give the information the article provides. Consider the ...
1
vote
1answer
21 views

Check my reasoning that this piecewise function is continuos

We have the function $f:D \rightarrow R$ defined on the domain $D = [0,1] \cup (2,3]$. The function is defined as $f(x) = \left\{\begin{matrix} x&0 \leq x \leq 1 \\ x-1 & 2 < x \leq 3 ...
-1
votes
1answer
23 views

Why does the sequence of functions have to be increasing/decreasing in Dini's theorem?

I think as long as a sequence of continuous functions converges pointwise to a continuous function on a closed interval, the convergence would be uniform. Can someone tell me why it's wrong?
0
votes
2answers
46 views

totally bounded question

Given: If (x,p) is totally bounded and E is subset of X, prove that (E,p) is totally bounded we know that Every subset of a totally bounded space is a totally bounded set; but even if a space is ...
0
votes
2answers
17 views

Application of Alternating series test

Is the series $\displaystyle\sum_{n=1}^\infty \frac{(-1)^n n^n}{n!e^n}$ convergent? This is inconclusive by ratio test. I tried to use root test, but ended up with ...
4
votes
3answers
48 views

Limit theorems, prove function has a limit at every point

Suppose that $f:R\to R$ is a function such that $f(x+y)=f(x)+f(y)$ for all $x,y∈R$. Assume that $f$ has a limit at $0$, $f(1)=1$. Prove that $f(x)=x$ for all $x \in R$ Hint: Show first that $f$ is ...
1
vote
3answers
55 views

If $f(1) > p >1$ then $f$ is increasing.

Consider the function $f$ with the following properties: $$\lim_{x\rightarrow 0} f(x) =1,$$ $$f(x+y)=f(x)\,f(y),$$ $$f(x) >0,\quad \forall x\in\mathbb{R},$$ $$ -\infty<x,y<\infty.$$ Show ...
0
votes
0answers
28 views

Let $f:R\to R$ and let $ C$ in $R$. [on hold]

Show that $$\lim_{x\rightarrow c} f(x) = L$$ if and only if $$\lim_{x\rightarrow 0} f(x+c)=L$$
0
votes
1answer
15 views

showing (0,1) with absolute value metric is not a complete metric space

According to the wikipedia, The open interval (0,1), with the absolute value metric, is not complete But i could not find any proof for this one Does anyone have any proof to show that (0,1) with ...
1
vote
1answer
27 views

Determine a condition on |x-1| that will assure that

$$|x^2-1|< 1/n$$ for a give n is a natural number. Here is my own solution, please check it for me: $$|x^2 -1|=|x+1||x-1| \leq (|x+1|)|x-1|.$$ $$ |x|=|x-1|+1.$$ we chose $$𝛿_1:=1$$ Then if $$ ...
0
votes
1answer
13 views

What is the average of two stochastic processes multiplied?

Consider two random processes $X(t)$ and $Y(t)$ for which $$\langle X(t) X(t') \rangle = \mu_X^2 + \sigma_X^2 \delta(t-t')$$ $$\langle Y(t) Y(t') \rangle = \mu_Y^2 + \sigma_Y^2 \delta(t-t')$$ ie. the ...
4
votes
1answer
33 views

How can I prove/disprove that the function $f(x)$ satifies $f'(c)=1$ for certain conditions

The function is defined on the interval $[0,1]$ with following conditions: 1) $f(0)=1$, 2) $f(1)=2$, 3) $f(x)$ is continuous on $[0,1]$, Prove or disprove: There exists some $c$ from $(0,1)$, such ...
1
vote
2answers
34 views

Show $f(rx) = [f(x)]^r$ where $r\in\mathbb{Q}$.

Consider the function $f$ with the following properties: $$(1) \lim_{x\rightarrow 0} f(x) =1,$$ $$(2) f(x+y)=f(x)f(y),$$ $$ -\infty<x,y<\infty.$$ Show that $f(rx)=[f(x)]^r$ where ...
1
vote
2answers
39 views

Showing $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}z^{k-1} = - \displaystyle\frac{\ln(1-z)}{z}$

Given the sum $\displaystyle\sum_{k=1}^{\infty}\frac{1}{k}z^{k-1} = 1 + \frac{z}{2} + \frac{z^2}{3} + ...$ How does one show that this sum is equal to $- \displaystyle\frac{\ln(1-z)}{z}$?