1
vote
1answer
32 views

Don't understand inequality in order to prove Algebraic Limit Theorem

I'm self-studying from the book Understanding Analysis by Stephen Abbott and I'm stuck on Theorem 2.3.3 on page 45, i.e., the Algebraic Limit Theorem. In particular, letting $\lim a_n = a$ and $\lim ...
0
votes
2answers
74 views

Proving uniform continuity of absolute value

Prove that the function $f(x) = |x-a| - |x-b|$ is uniformly continuous on $\mathbb{R}$.
0
votes
1answer
47 views

Equivalent form not using absolute values

Looking at the solution of Trench´s Introduction to Real Analysis exercises, I am struggling with this: Write the following expression in equivalent form not involving absolute values: $a + b + 2c + ...
0
votes
1answer
47 views

Periodic absolute value function

Define $$h(x)=|x|$$ on the interval $[-1,1]$ and extend the defintion of $h$ to all of $\mathbb{R}$ by requiring that $h(x+2)=h(x)$. Now define the function: $$h_n (x)=\frac{1}{2^n} h(2^n x)$$ ...
1
vote
4answers
124 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 ...
1
vote
1answer
44 views

Explanation for |x|=-x if x<0

Can someone explain $|x|=-x$ if $x<0$ . I've proven various theorems in my real analysis text for homework, but I cannot see how $|x|=-x$ if $x<0$ makes sense.
0
votes
1answer
46 views

Complex number in polar coordinates

I have to get $\Im$, $\Re$, the absolut value as well as the argument $\phi$ of the complex number $$z = \left(-\frac{1}{\sqrt2}+\sqrt\frac{3}{2}i\right)^8$$ I do this by transforming $z' = ...
3
votes
3answers
110 views

Proving continuity of a absolute value function

How can i prove the function $f: x \mapsto x|x|$ is continuous over $\mathbb{R}$ using epsilon-delta definition. I've tried: Given a certain $\epsilon$ we want to prove that there exists a $\delta$ ...
0
votes
1answer
59 views

Problem with absolute value

Say that $|\sqrt{x}-1| < \epsilon$. I am having a problem with handling this inequality. I want to exclude x. I. $|\sqrt{x}-1| < \epsilon$ $|\sqrt{x}| - |1| \leq |\sqrt{x}-1| < \epsilon$ ...
1
vote
2answers
52 views

Proof by contradiction: $c<a<d \wedge c<b<d \to |a-b|<d-c$

Let be $a,b,c,d \in \mathbb{R}$, I must proof "$c<a<d \wedge c<b<d \to |a-b|<d-c$". Proof by contradiction: I have $|a-b|\geq d-c$, therefore $a-b \leq c-d \vee a-b \geq d-c$ (or $a-c ...
2
votes
2answers
73 views

$|x|=\max\{-x,x\}=\max\{-x,x,0\}$?!

Let $x \in \mathbb{R}$, $|x|=\max\{-x,x\}$, is correct also $|x|=\max\{-x,0,x\}$? Thanks in advance!
5
votes
3answers
129 views

How would I prove $|x + y| \le |x| + |y|$?

How would I write a detailed structured proof for: for all real numbers $x$ and $y$, $|x + y| \le |x| + |y|$ I'm planning on breaking it up into four cases, where both $x,y < 0$, $x \ge 0$ ...
2
votes
4answers
141 views

Prove that $||x|-|y|| \leq |x-y|$ [duplicate]

$||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ In Principles of MA(Rudin), the author said one sees easily that $||x|-|y|| \leq |x-y|$ when $(x,y \in R^k)$ (p.88, Rudin) from the triangle ...
1
vote
4answers
280 views

Please help me to prove this inequality: $|x|+|y|≥|x+y|$

Please help me to prove the following inequality: $|x|+|y|\geq|x+y|$ in which $x$ and $y$ are real numbers. Any help or hint would be appreciated. Thanks :)
0
votes
3answers
60 views

finding values for absolute convergence

Find all values of real number p or which the series converges: $$\sum \limits_{k=2}^{\infty} \frac{1}{\sqrt{k} (k^{p} - 1)}$$ I tried using the root test and the ratio test, but I got stuck on ...
2
votes
2answers
90 views

Absolute Convergence of a Series

Find all values of real number p for which the series converges absolutely $$\sum_{k=2}^{\infty} \frac{1}{k\, (\log{k})^p}$$
0
votes
1answer
135 views

For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$?

I came across the following question: For what $ \alpha \in \mathbb R$ is $ |x|^\alpha $ differentiable in $x=0$? What I have tried: Since for $ \alpha = 1 $ is clearly non-differentiable in ...
2
votes
2answers
227 views

Why is the derivative of $\frac{|x|}{x}$ equal to $\emptyset$ at $x=0$?

I got a bit of a confusion here. If $\varphi(x)=\frac{|x|}{x}$, then $$ \varphi(x) = \left.\Bigg\{ \begin{array}{cc} 1 &if \ x>0\\ \emptyset & if \ x=0\\ -1 & if \ x <0 \end{array} ...
3
votes
1answer
104 views

Integral of absolute e-function

I have to integrate the following function: $$\int e^{-|x|}$$ I tried this and I don't think, that this is right. So can you tell me, where my fault is? $$\int e^{-|x|} = ...
5
votes
1answer
121 views

An absolute value problem

Let $a$ and $b$ in $\mathbb{R}$ 1) Show that $||a|-|b||\leq|a+b|\leq|a|+|b|$. 2) Prove that the one or the other of the two inequalities is an equality. It's fine whit the 1st question but i can't ...
0
votes
2answers
213 views

absolute value inequality limit

Could someone verify this proof? Prove $\lim\limits_{n\to\infty} C_n = \lim\limits_{n\to\infty} \dfrac{4n+3}{7n-5} = \dfrac{4}{7}$ Proof: Let $\epsilon > 0$ and take $N = \dfrac{41}{49\epsilon} + ...
3
votes
1answer
2k views

General Proof for the triangle inequality

I am trying to prove: $P(n): |x_1| + \cdots + |x_n| \leq |x_1 + \cdots +x_n|$ for all natural numbers $n$. The $x_i$ are real numbers. Base: Let $n =1$: we have $|x_1| \leq |x_1|$ which is clearly ...
1
vote
3answers
1k views

Proving absolute value inequalities

I need help proving the last two cases for the following inequality: $\bigl|\lvert x\rvert-\lvert y\rvert\bigr| \le \lvert x-y\rvert$. Case 1: $x > 0$ and $y > 0$: the inequality simplifies ...
0
votes
2answers
86 views

Two Analysis Questions

1) Define : $\langle z\rangle := (1+|z|^2 ) ^\frac{1}{2} $ for all $z \in \mathbb{C} $. Prove : $\langle x+y\rangle \leq 2\langle x\rangle\langle y\rangle $ for all $x,y \in \mathbb{R} ^N$ . 2) ...
2
votes
1answer
11k views

Reverse Triangle Inequality Proof

I've seen the full proof of the Triangle Inequality $|x+y|\le|x|+|y| $ However, I haven't seen the proof of the reverse triangle inequality: $||x|-|y||\le|x-y|$ Could you please prove this using ...
-3
votes
1answer
454 views

What properties of the absolute value function should one remember? [closed]

When one begins to study real analysis, the absolute value function quickly enters and a large number of exercises involve manipulations with it. What are the basic properties of absolute value that ...
5
votes
1answer
987 views

Prove variant of triangle inequality containing p-th power for 0 < p < 1

Sorry if this is a trivial question, but I am kind of stuck with proving the following inequality and have been searching for a while: $\rho \left( \sum\limits_i^n d_i \right) \leq \sum\limits_i^n ...
2
votes
1answer
131 views

$|x-y| + |y-z| = |x-z|$ then $x \le y \le z$

I'm doing this exercise from Robert Bartle's Introduction to Analysis, it's a if only if excersise and I've done the half part, but I can't figure this part of the ...