0
votes
2answers
21 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
35 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
30 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
0answers
8 views

prove where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable and is not differentiable [duplicate]

Classify where $f(x,y) = \sqrt{|x| + |y|}$ is differentiable -- where it's not, prove it, and where it is, prove it. For $x\neq 0$ and $y\neq 0$, we can just treat it as $f(a,b) = \sqrt{a+b}$ (where ...
0
votes
1answer
34 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
41 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
53 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
37 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
41 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
110 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
48 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. ...
3
votes
0answers
57 views

My first simple direct proof (very simple theorem on real numbers). Please mark/grade.

What do you think about my first simple direct proof? What mark/grade would you give me? Besides, I am curious about whether you like the style. Theorem Let $I = [a,b]$ be a non-empty closed ...
1
vote
1answer
38 views

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$.

show that for any $n \in \mathbb{Z}$ gcd($n^2 - n + 1, n +1)$ is either $1$ or $3$. My Work: I considered the case where $n =-1$ , and the case $n \not= 1$. So when $n\not= -1$ we can let $n^2 - ...
0
votes
3answers
25 views

Proving some number is a subsequential limit

Let $X_n$ be a sequence of real numbers. Suppose that for every $\epsilon>0$ and for every $m\in{N}$, there exists $n\geq m$ with $|x_n|<\epsilon$. Prove that 0 is a subsequential limit of the ...
1
vote
2answers
61 views

$\forall x \in \mathbb{R}$ show that $x=\sum_{n=1}^\infty k_na_n = \prod_{n=1}^{\infty}m_na_n$ …

Yet again, another cool problem from the book "problems in mathematical analysis" by Piotr & Witkowski: Prove that if $a_n \neq 0$, $n=1,2,\cdots$ and $\displaystyle \lim_{n \to \infty} a_n = 0$, ...
0
votes
1answer
58 views

Proving a Sequence Diverges

Let {${\alpha_n}$} be a sequence in $\mathbb R$ satisfying $|\alpha_n - \alpha_{n+1}|\geq c$ for some $c > 0$ and all $n \in \mathbb N$. Prove that the sequence {$\alpha_n$} diverges. I've looked ...
1
vote
3answers
54 views

Let $P(x)$ be any polynomial and suppose that $a_n \rightarrow a$. Prove $lim_{n\rightarrow\infty} P(a_n) = P(a)$

I know the limit rules but that's not helping me out much here this seems so simple but I don't even know where to start. I read the chapter on this and they don't do any examples like this. I ...
1
vote
4answers
104 views

Proving the inequality $|a-b| \leq |a-c| + |c-b|$ for real $a,b,c$

Let $a,b,c$ real numbers. Prove the inequality $|a-b| \leq |a-c| + |c-b|$. Prove that equality holds if and only if $a \leq c \leq b$ or $b \leq c \leq a$. I've tried starting with just $a \leq ...
0
votes
6answers
92 views

Suppose that $x$ and $y$ satisfy $\frac{x}{2} + \frac{y}{3} = 1$. Prove that $x^2 + y^2 > 1$.

Ok , i tried to prove this via Contrapositive setting $x^2 + y^2 \le 1$. After doing some algebra i have arrived at $x \le \sqrt{-y^2}$. I'm fairly sure this isn't right. I also solved for x and y in ...
1
vote
0answers
55 views

Don't choose, refrain from delta, epsilon, or $N \in \mathbb{N}$ in delta-epsilon proofs.

(http://math.stackexchange.com/a/700667/85079) I would write the proof with all my bounds $\eta$ and then choose $\eta$ to make the conclusion match the arbitrary $\epsilon$. ...
0
votes
2answers
70 views

Prove that there is a real solution of $x=e^{-x}$

I know I have to use the intermediate value theorem but how?
2
votes
2answers
55 views

Another Epsilon-N Limit Proof Question

How to prove the limit of the following sequence using epsilon-N argument. $$a_n=\frac{3n^2+2n+1}{2n^2+n}$$ I took the limit to be $\frac{3}{2}$ and proceeded with the argument, ...
0
votes
3answers
35 views

Help with proof of continuous functions with neighborhood value $N$.

Prove that if $f(x)$ and $g(x)$ are continuous at $c$ and $f(c) < g(c)$ then there is a neighborhood $N$ of $c$ such that $f(x) < g(x)$.
0
votes
2answers
50 views

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M

Prove that if M = supS for some nonempty bounded set S, then there exists an increasing sequence sn, of points S such that lim sn = M Is this a monotone sequence? Do I need to use Cantor's principle
0
votes
3answers
39 views

Proving the sandwich theorem for $\lim_{n \to \infty} c_n$ if $a_n \leq c_n \leq b_n$ and $a_n, b_n \to c$

Suppose $\lim\limits_{n \rightarrow \infty} a_n =\lim\limits_{n \rightarrow \infty} b_n = c$ and $a_n \le c_n \le b_n$ for all $n$. Prove that $\lim\limits_{n \rightarrow \infty} c_n = c$. How ...
0
votes
2answers
118 views

How to prove the equation |xy|=|x||y| if we assume x and y are real numbers by using analysis. [closed]

Prove that if x and y are real numbers, then |xy|=|x||y|. Hint check all the cases. I tried assuming the left hand side equals the right hand side if we remove absolute values. Also, tried using the ...
0
votes
1answer
46 views

Strange derivative

In this proof: http://www.math.hmc.edu/calculus/tutorials/mean_value/proof_mean.html Why does $g'(x) = f'(x) - \frac{f(b)-f(a)}{b-a}$?
1
vote
1answer
23 views

Proof using sets and infinimums

Let $S$ and $T$ be nonempty sets of real numbers, bounded below. Prove that $$\inf(S\cup T) = \min \{\inf S,\inf T \} $$ So the answer almost seems obvious here, I get that obviously the inf of the ...
2
votes
1answer
21 views

Question with nonempty bounds and sets

Let $A$ and $B$ be nonempty sets of real numbers, bounded above and below. Prove that if $A\cap B$ is also nonempty, then $infB\leq supA$. So my train of thought goes like this: I'm picturing that ...
1
vote
0answers
28 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 ...
0
votes
1answer
70 views

Proof: Lemma 5.6.2 - Elements of Real Analysis (C.G. Denlinger)

I reading the book "Elements of Real Analysis, C.G.Denlinger".. and I need the proof of Lemma 5.6.2: Thanks in advance!
0
votes
2answers
40 views

Contrapositive Proof: Specific Question! Need help!

I've been stuck on this question for a few days, please help me with this contra positive proof! Suppose that $x$ and $y$ satisfy $\frac 1 2 x + \frac 1 3 y = 1$. Prove that $x^2 + y^2 > ...
0
votes
0answers
24 views

Equivalence formulation of continuity

Proposition (Equivalence formulation of continuity): Let $X\subset \mathbb{R}, f:X \rightarrow \mathbb{R}, x_0\in X$. Then the following statements are logically equivalents. (1) For every sequence ...
-2
votes
2answers
42 views

Prove that $a_n$ = $2^n$ + $(-1)^n$

I am given the sequence $a_0= 2$, $a_1= 1$ and $a_{n+2}= a_{n+1} + 2a_n$ How do I prove that $a_n$ = $2^n + (-1)^n$
2
votes
1answer
43 views

Prove that if SupS = infinity then for every N > 0 there exists an element s of S such that s > N

How would I prove this? Would I use upper bounds or lower bounds or would I do a proof by contradiction?
3
votes
2answers
193 views

Stuck with a tricky existence proof

Show that there exists a continuous function $f: [-1, 1] \rightarrow \mathbb{R}$ such $f(0) = 1$ and $f(x) = \frac{2-x^2}{2} \cdot f(\frac{x^2}{2-x^2})$ $\forall x \in [-1, 1]$ I tried putting ...
2
votes
2answers
53 views

Upper bound of a set (equivalence problem)

I took a real analysis course three years ago and I unfortunately didn't get all of it, starting with basics. Question: Let $E$ be a set of real numbers. Show that $x$ is not an upper bound of $E$ ...
1
vote
1answer
62 views

How do i do this process *precisely*?

Let $[a,b)\times [c,d)$ be a rectangle $R$ in $\mathbb{R}^2$. Let $\{[u_k,v_k)\times [p_k,q_k)\}_{1≦k≦n}$ be a mutually disjoint finite sequence whose union is $R$. Then we can decompose this into ...
2
votes
0answers
91 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 ...
0
votes
1answer
67 views

Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that… [duplicate]

Let $a_1,a_2,\ldots,a_n$ be positive real numbers.Prove that $$\lim_{x\to 0}\bigg(\dfrac{a_1^x+a_2^x+\cdots+a_n^x}{n}\bigg)^{\frac{1}{x}}=\sqrt[n]{a_1a_2\cdots a_n}$$ Got no clue where to begin from ...
1
vote
1answer
60 views

Let $x$ be a real number. To prove…

Let $x$ be a real number. Define the sequence $(x_n)_{n\ge1}$ recursively by $x_1=1$ and $x_{n+1}=x^n+nx_n$ for $n\ge1$. Prove that, $$\prod_{n=1}^\infty \bigg(1-\dfrac{x^n}{x_{n+1}}\bigg)=e^{-x}$$ ...
1
vote
3answers
69 views

Show that $(x_n)$ is decreasing and find its limit.

Let $0<x_1<1$. For $n \in \mathbb{N}$, let $x_{n+1}=1- \sqrt{1-x_n}$. Show that $(x_n)$ is decreasing and find its limit. I did: $$x_{n+1} = 1- \sqrt{1-x_n}$$ $$x_{n+1} - x_n= 1- \sqrt{1-x_n} - ...
1
vote
3answers
129 views

Prove that $\sqrt[n]{n!}$ is increasing and diverges

I have to prove that $a_n$ is (strictly) increasing and diverges $a_n = \sqrt[n]{n!}$ ; n $\in$ $\ \mathbb{N}$ From sequence I see that $a_n$ increasing to infinitive. $\sqrt[1]{1!}=1 ,\ ...
3
votes
0answers
69 views

Interchange of finite sum with convergence sequences.

Hi everyone I'm wondering if the following proof is correct (to be honest at the beginning I have some troubles to understand what the sequences of the sums of convergent sequences has to be, but I ...
2
votes
1answer
196 views

Proving a constant function $f(x) = c$ is Riemann integrable

Prove that a constant function $f(x) = c$, where $c$ is in the Real Numbers, is Riemann integrable on any interval $[a, b]$ and $\int_a^bf(x) dx = c(b-a)$. By looking at the definition, it looks ...
2
votes
3answers
209 views

Proving if a function $f$ is differentiable and $f'(x)\ne0$ at all $x$, then it is one-to-one

Here's what the problem reads: Suppose that the function $f$ is continuous on $[a,b]$ and differentiable on $(a,b)$, and $f'(x)\neq 0$ for any $x \in (a,b)$. Prove that $f$ must be one-to-one. ...
1
vote
2answers
34 views

Limit and maximum: IVT

Let $f$ be a function defined and continuous on $\mathbb{R}$. Assume that $f(a) > 0$ for some $a \in \mathbb{R}$ and that $$\lim_{x\to \infty} f(x) = 0 = \lim_{x\to -\infty}f(x)$$ Show that ...
3
votes
0answers
74 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 ...
1
vote
2answers
68 views

If a function $x\mapsto xf(x)$ has a derivative at $a \ne0$, then $f$ is differentiable at $a$

Prove if $x\mapsto xf(x)$ has a derivative at $a \neq 0$, then $f$ is differentiable at $a$. The problem I'm encountering is with the setup. If I am trying to show $$\lim_{x \to ...
1
vote
3answers
83 views

Absolute Value Proof: if $-a \leq x \leq a$, then $|x| \leq a$.

I want to prove the following proposition: If $-a \leq x \leq a$, then $|x| \leq a$, where $x,a \in \mathbb{R}$. Here's my proof: By trichotomy, there are two possibilities: either $x \geq ...