4
votes
3answers
35 views

If d is a norm on V, is $\frac{d(x,y)}{1+d(x,y)}$ a norm on V?

Let d be a norm on a vector space V and let $\psi:V \to [0,\infty)$ be a function defined as $\psi(v)=\frac{d(v)}{1+d(v)}$. Is $\psi$ a norm on $V$? It seems that $\psi$ does not satisfy the ...
6
votes
0answers
115 views

Finitely additive function on an infinite set, s.t., $m(A)=0$ for any finite set and $m(X)=1$ (constructive approach)

Other exercise which I found in Dudley's Analysis book: Show that there is a measure on a infinite set $X$, defined on $2^X$ s.t. is finitely additive, $m(A)=0$ for any finite set and $m(X)=1$. ...
2
votes
4answers
81 views

Prove that a continuous function defined on an interval $[a,b]$ has a fixed point.

I have to prove that : Suppose that $f:[a,b] \to [a,b]$ is continuous. Prove that there is at least one fixed point in $[a,b]$. But I don't know how to attack it since I can't apply anything of ...
2
votes
4answers
100 views

Constructive proof for existence of integer part of real number

I try to prove de following exercise of my analysis textbook. Show that for every real number $x$ there is exactly one integer $N$ such that $N \le x < N + 1$. I have been finding a ...
0
votes
1answer
59 views

Two reasons why $\int^{1}_{0}f(x) \,dx$ exists?

Consider $f$ on $[0,1]$ defined as $f(0)=0$ $$f(x)=2^{-n}\quad \text{if}\quad 2^{-n-1}<x\le2^{-n},$$ for $n=0,1,2,3,...$ I'm looking for two reasons why $\int^{1}_{0}f(x) \,dx$ exists? One ...
3
votes
2answers
82 views

square root of 2 irrational - alternative proof

I have found the following alternative proof online. It looks amazingly elegant but I wonder if it is correct. I mean: should it not state that $(\sqrt{2}-1)\cdot k \in \mathbb{N}$ to be able to ...
1
vote
0answers
60 views

Measure of Elementary Sets Proof

I am struggling with what seems like a very simple problem from Terrence Tao's Introduction to Measure Theory book (which is available for free online by the way). What I am trying to prove is the ...
4
votes
0answers
90 views

Is it possible to find $\int \frac{1}{\sqrt[4]{1+x^4}} dx$ by parametrizing the curve $y^4-x^4=1$?

I found this integral in a handbook of integrals: $$\int \frac{1}{\sqrt[4]{1+x^4}} dx$$ I already have evaluated this integral by trigonometric substitutions and my answer agrees with that of the ...
0
votes
3answers
114 views

Prove $\lim\limits_{x\to 0^+} \frac{\ln x}{x} = -\infty$

Prove $\lim\limits_{x\to 0^+} \frac{\ln x}{x} = -\infty$ I've seen the following proof but I think it's invalid: $$\lim\limits_{x\to 0^+} \frac{\ln x}{x} = \lim\limits_{x\to 0^+}\ln x \cdot ...
1
vote
1answer
59 views

Alternate proof for a theorem on ordered fields

I came across the following theorem, while studying "A First Course in Real Analysis" by Berberian Sterling. In an ordered field, if $a, b, c \geq 0$ and $a \leq b+c$, then $$ {{a}\over{1+a}} \leq ...
1
vote
1answer
95 views

Check proof please

Prove that if: $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ than: $\lim_{x\rightarrow\infty}{\frac{f(x)}{x}}=L$ Assuming $\lim_{x\rightarrow\infty}{f(x+1)-f(x)}=L$ we can choose $X_{\epsilon}$ s.t. ...
2
votes
3answers
92 views

If $I_{n+1}\subset I_n$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty

Question: If $I_n$ is closed and bounded, $I_{n+1}\subset I_n$, and $I_n\neq\emptyset$, show that $\bigcap_{n=1}^\infty I_n$ is nonempty. This is not a homework help question. I'm actually looking ...
2
votes
3answers
84 views

$x^{1+\epsilon}$ is not uniformly continuous on $[0,\infty)$

There are two questions. First: is the proof underneath correct? Let $\epsilon>0$ and let $f(x)=x^{1+\epsilon}$. I aim to show that $f$ is not uniformly continuous on $[0,\infty)$. We will show ...
0
votes
2answers
70 views

Proof that doesn't exists a rational $s$ such that $s^2 = 6$

Well, I solved it, and I would like to know if there is anything that can be corrected or improved here. I think that the proof ended up too long, and with too many letters. Surely there is a better ...
0
votes
1answer
37 views

Convergence of Alternating harmonic series (Direct!)

Once again, note No use of the alternating series test!
15
votes
2answers
456 views

A snappy proof of Fatou's lemma

I'm studying basic real analysis and stumbled across three big results (Fatou's lemma, Lebesgue's Dominated Convergence theorem, and the Monotone Convergence theorem) in the theory of Lebesgue ...
1
vote
2answers
96 views

Proof for $\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$ without complexes? [duplicate]

This is what I needed. Practically, a link were also okay. $$\sum_{n=1}^{\infty}\frac{1}{n^2}=\frac{\pi^2}{6}$$
6
votes
3answers
257 views

Proving $fg$ and $f+g$ is Riemann integrable through the easy and hard way.

Problem: Suppose $f,g$ are Riemann integrable functions, show that $f+g$ and $fg$ are also Riemann integrable. I know there is really easy to do this with measure theory, but I want to see if ...
1
vote
3answers
65 views

Sequence that contains subsequences converging to every point in the infinite set $\{1/n \} \, \forall \, n \in N$ (Abbott p 58 q2.5.3c)

has this property. Notice that there is also a subsequence converging to 0. We shall see that this is unavoidable. I acquiesce to this example, but I wasn't conscious of it until I read ...
0
votes
1answer
56 views

$x_n,x_ny_n$ convergence implies $y_n$ converges

Assume that $x_n$ converges to a nonzero number $x$ and that the sum $x_ny_n$ converges to a limit $L$. Prove that the series $y_n$ converges. The natural guess is that $y_n$ will converge to $L/x$. ...
0
votes
1answer
59 views

Abel's test: how to prove it

Consider the following statement: Let $b_n$ satisfy $b_1 \ge b_2 \ge \dots \ge 0$ and let $\sum_{n=1}^\infty a_n$ be a series for which the partial sums are bounded i.e. there exists $A > 0$ such ...
1
vote
1answer
75 views

Proof without mean value theorem

Is it possible to prove the following without using the mean value theorem: If $f$ is differentiable on an interval containing $0$ and if $\lim_{x \to 0} f'(x) = L$ then $f'(0) = L$. I have ...
2
votes
2answers
100 views

How to prove from the definition that $X_n \xrightarrow{\mathbb{P}} X$ implies $\frac{1}{X_n} \xrightarrow{\mathbb{P}} \frac{1}{X}$?

Let $X, X_1, X_2, \ldots : \Omega \to (0,\infty)$ be random variables such that $X_n \xrightarrow{\mathbb{P}} X$. I'd like to show from the definition that $\frac{1}{X_n} \xrightarrow{\mathbb{P}} ...
10
votes
4answers
355 views

How to prove $f\equiv 0$ without Weierstrass theorem?

Let $\,f:[0,1] \to \mathbb{R}$ continuous. Show that: If $$\int_0 ^1 x^k f(x)\, dx=0,$$ for all $k\in\mathbb N$, then $f\equiv 0$. I know that it can be proved using Weierstrass Theorem, ...
1
vote
0answers
36 views

Metric space, I would like to rewrite this proof, completeness and compactness

I am looking at metric spaces. For the metric space C(X,Y) the metric is: $\rho (f,g)=sup\{|f(x)-g(x)| :x \in X\}$ This is the proof I am reading about. I would like to do the proof without just ...
2
votes
0answers
96 views

The Heine-Borel Theorem for the real line

Hi everyone I'd like to know if the following argument is correct and also I'm very interested in a constructive approach for (2)$\Rightarrow$(1) (a link or a hint it will sufficient for me) I was ...
7
votes
1answer
358 views

Uniform Convergence verification for Sequence of functions - NBHM

Following is a list of problems from an exam for admission into Ph.D program. I have just compiled all previous questions on uniform convergence of sequence of functions and i tried to work out . I ...
1
vote
1answer
61 views

Prove that $ \lim (s_n t_n) =0$ given $\vert t_n \vert \leq M $ and $ \lim (s_n) = 0$

Let $ (t_n) $ be a bounded sequence, i.e., there exists $ M $ such that $ \vert t_n \vert \leq M $ for all $ n $, and let $ (s_n) $ be a sequence such that $ \lim s_n = 0 $. Prove $ \lim (s_n t_n) ...
2
votes
4answers
186 views

Prove that $ \lim_{n \rightarrow \infty } \frac{n+6}{n^2-6} = 0 $.

My attempt: We prove that $ \lim\limits_{n \rightarrow \infty } \dfrac{n+6}{n^2-6} =0$. It is sufficient to show that for any $ \epsilon \in\textbf{R}^+ $, there exists an $ K \in \textbf{R}$ such ...
4
votes
3answers
210 views

Prove that $ \displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $.

My attempt: We prove that $$\displaystyle \lim\limits_{n \rightarrow \infty} \left(\frac{23n+2}{4n+1}\right) = \frac{23}{4} $$ It is sufficient to show that for an arbitrary real number ...
2
votes
0answers
104 views

Absolute Convergent Series Laws (Exercise)

Me again, I have troubles to understand the concept of absolute convergence in the general setting when the set can be uncountable. In the book what I read says: Definition: Let $X$ be a set ...
2
votes
2answers
189 views

Is $f_n(x)=n^2x(1-x^2)^n$ uniformly convergent on $[0,1]$?

Does $f_n(x)=n^2x(1-x^2)^n$ converges uniformly on $[0,1]$ $\lim_{n\to \infty} f_n(x) =f(x)=0$ and then I calculated sup of $|f_n(x)-f(x)|$ which came out to be $\frac{n^2}{\sqrt{2n+1}}\cdot ...
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 ...
3
votes
0answers
80 views

A question on mathematical writing.

One of the problems I am grading this week is as follows: Given a simply connected bounded domain $\Omega$ on $\mathbb{R}^{2}$, prove that there exist a line that separates it into two parts of equal ...
2
votes
1answer
71 views

The sequence $H_n-\ln(n)$ converges

Is there a proof that the sequence $\displaystyle \sum_{k=1}^n \frac{1}{k}-\ln(n)$ converges that doesn't use integrals?
0
votes
1answer
75 views

Starting Bisection Proof of Extreme Value Theorem

I am having difficulties beginning a proof for the following statement: Use a proof strategy of bisection to prove that every function $f:[a,b] \to \mathbb{R}$ that is not bounded above is ...
3
votes
1answer
139 views

Proofs regarding Continuous functions 2

I need verification for this proof: Q: Suppose $f: (0,1)\rightarrow \mathbb{R}$ is defined by $f(x) = \begin{cases}\frac{1}{n} & \text{if }\text{x is rational with x} = \frac{m}{n}\text{ in ...
1
vote
1answer
85 views

Proofs regarding Continuous functions 1

Q: Let $f: \mathbb{R}\rightarrow \mathbb{R}$ be a bounded function (that is, there exists some $M\geq 0$ so that $|f(x)|\leq M$ for all $x\in\mathbb{R}$). Define a new function ...
0
votes
1answer
148 views

Alternatinve proof for the principle of the Iterated Suprema

The back of the book gave a proof similar to the proof here Proving principle of the Iterated Suprema, but I proved it following way before I checked the back of the book. Could some one verify this ...
0
votes
2answers
68 views

prove there exists $x$ in ${\mathbb R}$ using the completeness axioms

Let $a, b \in {\mathbb R}$ with $a < b$. Prove that there exists $x\in {\mathbb R}$ which is NOT a rational number such that $a < x < b$. This is what i have at the moment.. It does not have ...
0
votes
1answer
110 views

Showing that $a_n \not \to 17$ implies a subsequence $a_{n_k}$ that is $\epsilon$ far from $17$ for some $\epsilon > 0$

I want to check my proof for this question: Suppose a sequence {$a_n$} does not converge to 17. Prove that there exists some $\epsilon$ > 0 and a subsequence {$a_{nk}$} so that $|a_{nk} - 17|$ > ...
3
votes
1answer
153 views

Help with a lemma of the nth root (without the binomial formula)

I have no idea of how to solve it. I would appreciate if someone gives me a hint, please. Definitions Let $\,x^{1/n}:= sup\{\, y \in \mathbb{R}: y\ge0 \text{ and } y^n\le x\, \}$ Lemma: Let ...
2
votes
2answers
213 views

Verification of proof of the Sequence of Arithmetic Theorem

Suppose $\left\{b_{n}\right\}$ is a sequence of real numbers which converges to $M$, so that $b_{n} \neq 0$ for each $n$, and $M \neq 0$. Prove that the sequence $\{ \frac{1}{b_n} \}$ converges to ...
2
votes
2answers
3k views

Prove that $\sup(A+B) = \sup(A) + \sup(B)$ and why does $\sup(A+B)$ exist?

We want to show that $\sup(A)+\sup(B)$ is the least upper bound of the set $A + B$. First, we need to show that $\sup(A) + \sup(B)$ is an upper bound for the set $A + B$. Indeed, if $z\in A + B$, then ...
2
votes
1answer
291 views

Proof for $-\sup(A) = \inf(-A)$

Let $A$ and $-A = \{ -x \mid x \in A \}$ be two bounded sets. I have to prove that $-\sup(A) = \inf(-A)$, i did it in the following way and wish to know if it is sufficient: $ \exists x\in A$ such ...
1
vote
1answer
91 views

How to improve this proof about real numbers?

I've trying to prove the following: let $a \in \mathbb{R}$ with $a \geq 0$ if for every real $\epsilon > 0$ we have $0 \leq a < \epsilon$ then $a =0$. I'm sure that there is a proof much ...
3
votes
2answers
264 views

Fundamental theorem of algebra: a proof for undergrads?

The fundamental theorem of algebra is the statement that a complex polynomial of positive degree has at least one root. I do not know complex analysis but I searched for proofs of the statement and ...
0
votes
4answers
2k views

Prove that a continuous function on a closed interval attains a maximum

As the title indicates, I'd like to prove the following: If $f:\mathbb R\to\mathbb R$ is a continuous function on $[a,b]$, then $f$ attains its maximum. Now, I do have a working proof: $[a,b]$ ...
5
votes
4answers
206 views

Prove the following identity

I am having some trouble proving following identity without use of induction, with which it is trivial. $$\sum_{n=1}^{m}\frac{1}{n(n+1)(n+2)}=\frac{1}{4}-\frac{1}{2(m+1)(m+2)}$$ I did expand the ...
2
votes
2answers
1k views

How can you prove the Nested Interval(s) Theorem in $\mathbb R$ without reference to the least upper bound property?

Claim: a nested sequence of closed bounded intervals in $\mathbb R$ has nonempty intersection. A textbook provides a proof using the least upper bound property of the real numbers, but adds an aside ...