0
votes
4answers
20 views

Why a sequence $\{x_k\}$ in $S$ such that $||x||>k$ cannot be a convergent subsequence?

This is an excerpt from my text: A set $S \subset \mathbb{R}^n$ is said to be compact if for all sequences of points $\{x_k\}$ such that $x_k\in S$ for each $k$, there exists a subsequence ...
3
votes
1answer
22 views

Help explain existence of a limit point of a sequence implies infinitely many $m$ where $d(x,x_m)<\epsilon$

I don't understand the phrase "...all but finitely many elements...". What does this mean exactly and how does the conclusion "Infinitely many elementsof the sequence $\{x_k\}$ must also be within ...
1
vote
2answers
32 views

Help with a sequence proof problem

I have the following theorem to prove, and the book makes a certain suggestion that I don't understand. Theorem Suppose that the sequence $\{a_{n}\}$ converges to $l$ and that the sequence ...
1
vote
1answer
44 views

A version of the Fundamental Theorem of Calculus for two variables

Let $f(x,y)$ be differentiable in the rectangle $R=[a,b]\times[c,d]$, show that the function $\displaystyle F(x,y)=\int_{a}^{x} f(t,y) \, dt$ is also differentiable in $R$ and that ...
1
vote
1answer
122 views

Proof that for every $\epsilon$ > 0 , there exists two rational numbers $q$ and $q'$ such that $q<x<q'$ and $\left |q-q' \right |< \epsilon$

I'm asked to prove that for every $\varepsilon$ > 0 , there exists two rational numbers $q$ and $q'$ such that $q<x<q'$ and $\left |q-q' \right |<\varepsilon$ where $x$ is a real number. ...
1
vote
3answers
27 views

Show that for all real numbers a and b, ab <= (1/2)(a^2+b^2)

so as in the title, I have the following theorem to prove. Theorem Show that for all $a$, $b$ $\epsilon$ $R$, that the following inequality holds, $\begin{equation} ab \leq \frac{1}{2}(a^2 + b^2) ...
0
votes
1answer
37 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 ...
0
votes
2answers
53 views

Using induction to prove a formula for $\sin x+\sin 3x+\dots+\sin (2n-1)x$

I'm working from the text "Intro To Real Analysis" by William Trench. Here is what I have thus far. I will prove using Mathematical Induction that $\sin x+\sin 3x+...+\sin (2n-1)x=\frac{1-\cos ...
0
votes
0answers
36 views

Correctness of proof based on nested interval theorem

I am given that for positive integer n, let $a_n=1-(1/n)$ and $b_n=1+(1/n)$. $I_n=[a_n,b_n]$. I am to show that 1) the hypothesis of the nested interval theorem are satisfied, 2) find the point of ...
1
vote
1answer
74 views

Proof that arithmetic and geometric mean converge

I need some help with understanding a part of this proof and also writing it up correctly. Given $a_n\geq a_{n+1}\geq b_{n+1} \geq b_n$ with $a_1=a$ and $b_1=b$. I am also given that ...
1
vote
3answers
120 views

Can the proof of Theorem 1.20 (b) in the book, The Principles of Mathematical Analysis by Walter Rudin, 3rd ed., be improved?

I'm reading Walter Rudin's PRINCIPLES OF MATHEMATICAL ANALYSIS, third edition, and am at Theorem 1.20(b), where he states and proves that between any to real numbers, there is a rational; that is, if ...
2
votes
2answers
45 views

How to prove that no constant can bound the function f(x) = x

I know this is a trivial question, but how would one mathematically demonstrate this using a proof?
1
vote
1answer
85 views

Book recommendations for these types of math?

I'm planning to write a math olympiad in a couple of months (4-5), and am just really trying to get the preparation in. I'm a fairly good math student (did ok in math, not an A+, but I got an A so my ...
0
votes
2answers
48 views

Proving the arithmetic mean equals the geometric mean when $a=b$.

Arithmetic mean $a,b \in \mathbb R$ is $A(a,b)=\frac{a+b}{2}$ Geomtric mean $a,b \in\left[0,\infty\right]$ is $G(a,b)=\sqrt{ab}$ I'm trying to prove that $G(a,b)=A(a,b)$ if and only if $a=b$. ...
0
votes
2answers
68 views

Prove intersection of open balls is another open ball

I was wondering how I would prove that an intersection of two open balls is also another open ball. The definition I have of an open ball is: If x $\in X$ and $\epsilon > 0$, $B_{\epsilon}(x) :=$ ...
2
votes
4answers
77 views

Is this a valid proof for the existence of a rational number between any two real numbers?

Given $a, b \in \mathbb R$ with $a<b$, prove that there exists some $r \in \mathbb Q$ such that $a<r<b$. Before I prove the main statement, there's a lemma I'd like to prove: Lemma ...
2
votes
2answers
59 views

Surjectivity of a piecewise function $f:(-1,1)\to \mathbb R$

Function $f$ is defined as $f: (-1, 1) \to \mathbb{R}$. $$ f(x) = \begin{cases} -x/(x-1),&x\geq 0 \\ x/(x+1),&x \leq 0 \end{cases} $$ Let $y \in \mathbb{R}$. How would I prove that there ...
0
votes
1answer
56 views

Prove that a function is continuous at x =0

I need to prove that $f$ continuous at $(x)=0$ using a $\epsilon$- proof $$ f(x) = \begin{cases} x/(1-x),&x\geq 0 \\ x/(1+x),&x \leq 0 \end{cases} $$ So this is what I have so far: Let ...
0
votes
0answers
46 views

Prove that a defined function g is continuous for a certain point

Consider the set of rational numbers $\mathbb{Q}$ equipped with the subspace topology inherited from $\mathbb{R}$. Let $c \in \mathbb{R}$. Define the function $g_{c}: \mathbb{Q} \to \mathbb{Q}$ via: ...
0
votes
1answer
69 views

A set $A \subseteq \mathbb{R}$ is closed if and only if every convergent sequence in $\mathbb{R}$ completely contained in A has its limit in A

Real analysis is a topic I'm unfamiliar with and I'm confused on how to write proofs on them. In order to prove that: A set $A \subseteq \mathbb{R}$ is closed (1) $\iff$ Every convergent sequence in ...
2
votes
2answers
75 views

Proving Lebesgue integration result

I have a Lebesgue integration question and a proposed proof. Please advise. Let $\Omega \subset \mathbb{R}^{n}$(denote the boundary as $\partial \Omega$) and consider $$\int_{\partial \Omega} vf ...
0
votes
3answers
110 views

Proving Riemann Sums via Analysis

Exercise $\bf 5.1.7$: Suppose $f:[a,b]\to\Bbb R$ is Riemann integrable. Let $\epsilon\gt0$ be given. Then show that there exists a partition $P=\{x_0,x_1,\ldots,x_n\}$ such that if we pick any set ...
2
votes
1answer
88 views

Euler proof of the formula involving factorial?

Let me be formal and write the formula Euler's Formula: Let $a$ and $n$ be nonnegative integers with $a\geq n.$ Then $$n!=a^n-\binom{n}{1}(a-1)^n+\binom{n}{2}(a-2)^n-\binom{n}{3}(a-3)^n+ > ...
0
votes
2answers
68 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
2answers
75 views

Showing convergence rigourously

So I have $f_n(x) = x^{4n} + \frac1{n^2}$ which I know converges to $f(x)=0$ uniformly on interval $[0,1)$, but how can I show this with rigour? Is this acceptably rigourous? $\lim ...
1
vote
4answers
34 views

Lowest possible value of a function with derivative greater than 2

I have the following two problems, and want to attempt to solve them with Mathematical rigour(which I don't yet possess): Suppose that $f$ is differentiable on $[1,4]$ and is such that $f(1) = 10$ ...
0
votes
0answers
28 views

U-substitution proof by partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
1
vote
2answers
41 views

Show a double-sided infinite integral of $\sin(x+b)$ exists iff $b=n\pi$

More formally: Show that $$\lim_{a\rightarrow \infty} \int_{-a}^a \sin(x+b)$$ exists if and only if $b=n\pi$ for some $n \in \mathbb{Z}$. I get the intuition fine. The function is just a horizontal ...
4
votes
0answers
79 views

Proving u-substitution the hard way — use only definition of integration with partitions

I am trying to prove integration by substitution, i.e. for $f:[c,d] \rightarrow \mathbb{R}$ continuous and $\phi: [c,d] \rightarrow [a,b]$ continuous on $[c,d]$ and $C^1$ on $(c,d)$. Then define ...
4
votes
3answers
104 views

Existence of increasing sequence $\{x_n\}\subset S$ with $\lim_{n\to\infty}x_n=\sup S$

Let $S\subset\mathbb{R}$ be a non-empty bounded above set. Then there exists a monotone increasing sequence $\{x_n\}\subset S$ such that $$\lim_{n\to\infty}x_n=\sup S.$$ I'm struggling with ...
1
vote
0answers
37 views

Proof Validation Function From Integers to Rationals is Continuous

I am teaching myself real analysis, so any help is greatly appreciated. Let the function be defined as $F : Z \rightarrow Q$ where $Z$ is the set of integers and $Q$ is the set of rational numbers, ...
3
votes
1answer
51 views

$f'$ strictly increases and $f'(c)=0$. There exist $x_1 < c < x_2$ such that $f'(c)=\frac{f(x_2)-f(x_1)}{x_2 - x_1}$

Question: Let $f$ be continuous on $[a,b]$ and differentiable on $(a,b)$. Assume that $f'$ is strictly increasing. Show that for any $c\in(a,b)$ such that $f'(c)=0$, there exist $x_1, x_2 \in [a,b], ...
2
votes
0answers
63 views

Proof regarding derivatives and Mean Value Theorem.

Original question: $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$. Show that for any $c \in (a,b)$ that is not a point of maximum or minimum for $f'$, there exist $x_1, x_2 \in (a,b)$ ...
3
votes
1answer
64 views

$f$ is differentiable. If $\lim_{x \to c}f'(x)$ exists, then this limit must be $f'(c)$.

Please prove: Let $f:(a,b) \to R$ be differentiable function, and let $c \in (a,b). $ If $\lim_{x \to c}f'(x)$ exists and is finite, then this limit must be $f'(c)$. I tried doing it directly but ...
0
votes
0answers
32 views

Show if this is integrable (defined 1 on rationals, 0 else)

Define $f: [0,1] \rightarrow \mathbb{R}$ as $f(x) = \begin{cases} 0 & x \in \mathbb{Q} \\ 1 & x \notin \mathbb{Q} \end{cases}$ Find $\underline{\int_0^1f}$ and $\overline{\int_0^1f}$. Is ...
1
vote
1answer
25 views

Inequalities involving x and y.

I am asked to prove: $(x-y)^3 \ge x^3-3x^2y$ where $x,y$ are real and $0 < y < x$ I am told Bernoulli's inequality may help. I have however reduced this to $3xy^2 - y^3 \ge 0$. I have ...
0
votes
1answer
47 views

Help with proof for solving an ODE using contraction mapping theorem

I'm trying to follow a proof for solving the ODE $$\frac{df}{dx} = (f(x)+x)x$$ For $0 \leq x \leq 1$ with the initial condition $ f(0)=0$. The proof I am following goes like this Define ...
6
votes
1answer
124 views

Rudin's 'Principle of Mathematical Analysis' Exercise 3.14

Since I'm studying real analysis using this book by myself, I'm not sure whether or not my method to prove convergence of sequence is right. I'm working on the above question's (d), and my solution ...
2
votes
2answers
33 views

Proving a theorem about the limit of a function

The theorem is as follows: $$\exists L\in\mathbb R\ \bigg(\lim_{x \to c}f(x)=L\bigg)\iff\forall\varepsilon>0\ \exists\delta>0\ \forall x_1,x_2\in B_{\delta}(c):|f(x_1)-f(x_2)|<\varepsilon$$ ...
0
votes
2answers
31 views

Show that $f$ is everywhere differentiable and the partials commute

Take the function $$ f(x,y) = \begin{cases}\frac{x^3y -xy^3}{x^2+y^2} & (x,y) \neq (0,0) \\ 0 & (x,y) = (0,0) \end{cases}. $$ Show that it is everywhere differentiable and that $D_{1,2}f(0,0)$ ...
0
votes
2answers
33 views

How to quantify this statement

How do I quantify this statement? Let $x \in \mathbb{R}$. $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. When I am trying to prove this, I am led (in the course of ...
1
vote
1answer
41 views

Is there a more concise way?

I am currently trying to prove Taylor's Theorem, this is the proof I am given: I am finding this very long and hard to follow, can anyone give me a more concise, clear proof? Thanks
2
votes
1answer
47 views

Showing that if derivative is 0, function is constant ($f: U \rightarrow \mathbb{R}$ where $U \subset \mathbb{R}^n$)

Here's the question: Suppose that $f: U \rightarrow \mathbb{R}$ is differentiable on the open subset $U\subset \mathbb{R}^n$, and $Df(x) =0$ for all $x\in U$. Show that $f$ is constant on $U$. My ...
0
votes
1answer
43 views

Suppose that the sequence of prices{$p_k$} converges to a limiting price $\bar p$. What must $\bar p$ be?

We let $Q_k$ denote the supply of commodity, $D_k$ the demand for the commodity, and $p_k$ the price at $k$-th time. The demand depends on the current price, $D_k = a + b p_k$ and the supply depends ...
0
votes
1answer
47 views

General conceptual confusion relating to vacuous proofs and quantifier help

I need to prove the statement: Let $x \in \mathbb{R}$. Prove that $1 \le x \le 2$ if and only if $1 \le x \le 1+ 1/n$ for some $n \in \mathbb{N}$. So I start with the forward implication: If $1 ≤ ...
0
votes
1answer
54 views

When proving things how do I defer choosing a values?

The best example is what I've just tried to prove. Usually I do these proofs in 2 or 3 passes, or draw a margin to separate notes. In the example I want to defer picking an $\epsilon_f$ and ...
1
vote
0answers
44 views

Show that this is not differentiable at any point in $\mathbb{R}$

Define $\phi: \ \mathbb{R} \rightarrow \mathbb{R}$ by $$ \phi(x) = \begin{cases} x\ :\ 0\le x\le \frac{1}{2}\\ 1-x :\ \frac{1}{2} \le x \le 1 \end{cases}$$ And then extend ...
0
votes
1answer
61 views

Prove the least upper bound property using Bolzano Weierstrass theorem

Prove the least upper bound property using Bolzano Weierstrass theorem. I know there are quite a fair number of similar questions on the site, but none of them provide satisfactory proofs. Does ...
2
votes
1answer
221 views

Proof concerning equivalent definition of supremum using limits

I believe I have a proof to the following theorem: Let $S$ be a subset of $\mathbb{R}$ that is non-empty and bounded above. $s \in \mathbb{R}$ is the supremum iff $s$ is an upper bound of $S$ and for ...
0
votes
0answers
59 views

Let {$p_n$} be a sequence of points in the $\mathbb{R}^2$. Use the notion of convergence to solve the following

A) Define what it means for a point p $\in$ $\mathbb{R}^2$ to be a limit point of {$p_n$}. B) Prove that p is a limit point of {$p_n$} if and only if {$p_n$} has a subsequence which converges to p. ...