What is Inequalites: Definition and 22 Discussions

I am reading the following passage in these lecture notes (chapter 10, in the proof of theorem 10.3) on power series (and have seen similar statements in other texts): I'm confused about ##|x_0|<R##. If ##M=\sup (A)##, then for every ##M'<M##, there exists an ##x\in A## such that ##x>M'##...

The problem itself is easy. My question is regarding the proper use of inequality symbols. I only need to do the first part to show where I am having the issue. The forces I need to consider are the coaster car's weight ##W = mg## and the reaction ##A## of the tracks acting on it. With the...

Let ##\Omega## here be ##\Omega=\sqrt{-u}##, in which it is not difficult to realize that ##\Omega ## is real if ##u<0##; imaginary, if ##u>0##. Now, suppose further that ##u=(a-b)^2## with ##a<0## and ##b>0## real numbers. Bearing this in mind, I want to demonstrate that ##\Omega## is real. To...

The assignment says proof by induction is possible, I cannot figure out how this is supposed to work out. Does somebody know the name of this by any chance? Seeing a derivation might help come up with an idea for a proof. Thank you everybody.

When I learned about Catalan's conjecture, I morphed the expression ##3^2-2^3## to ##a^b+b^a##. Then, out of curiosity, I compared ##a^b+b^a## to ##a^a+b^b##. I have tried many numbers (greater that or equal to 1) for both ##a## and ##b##, noticing that ##a^a+b^b## is bigger than ##a^b+b^a##. I...

Let’s say we are given two inequalities $$ x \lt y \\ a \lt b$$ then we can write (we can even prove it using logarithms) $$ ax \lt by$$ given that every number is positive. In this article (Fact 4.1, point (ii) ) it is given that if ##x, y## are positive numbers and...

Homework Statement I'm currently working through Spivak independently and have reached the problems at the end of ch. 1. The problem is: Prove that if 0 < a < b , then a < \sqrt{ab} < \frac{a+b}{2} < b Homework Equations Spivak's properties P1 - P12 The Attempt at a Solution I was...

Homework Statement Prove that ##\forall n \in \mathbb{N}## $$\frac{n}{2} < 1 + \frac{1}{2} + \frac{1}{3} + \ldots + \frac{1}{2^n - 1} \leq n \text{ .}$$ Homework Equations Peano axioms and field axioms for real numbers. The Attempt at a Solution Okay so my first assumption was that this part...

Homework Statement Let ##S\subseteq \Bbb{R}## and ##T = \{ t\in \Bbb{R} : \exists s\in S, \vert t-s\vert \lt \epsilon\}## where ##\epsilon## is fixed. I need to show T is an open set. Homework Equations n/a The Attempt at a Solution Let ##x \in T##, then ##\exists \sigma \in S## such that ##x...

So, I know that the inequality √f(x)<g(x) is equivalent to f(x)≥0 ∧ g(x)> 0 ∧ f(x)<(g(x))^2. However, why does g(x) have to be greater and not greater or equal to zero? Is it because for some x, f(x) = g(x)=0, and then > wouldn't hold? Doesn't f(x)<(g(x))^2 make sure that f(x) will not be...

Homework Statement [/B] 2. The attempt at a solution I'm not really sure where to start. We just want to show that ##\lim_{x \to c} \frac{f(x) - f(c)}{x - c} = 0##. I see that ##\lim_{x \to c} (x - c)^2 = 0##. I feel that this may be a simple trick of inequalities, but I am having a complete...

Homework Statement [/B] In the equation between (3) and (2), why does the author says that ? Isn't the trigonometric identity actually ? 2. Homework Equations The Attempt at a Solution

An absolute value property is $$\lvert a \rvert \geq b \iff a\leq-b \quad \text{ or } \quad a\geq b,$$ for ##b>0##. Is this true for the case ##a=0##? I mean if ##a=0, \lvert a \rvert =0## so ##0 \geq b##. But ##b## is supposed to be ##b>0##, so we have a contradiction. How can this property...

Homework Statement Given : (y+2)(y-3) <= 0Homework EquationsThe Attempt at a Solution Now, I have y-3 <= 0 or y+2 <= 0 Hence, y <= 3 or y <= -2 But how is correct? I think is wrong because y <= -2. Can someone please clarify?

Homework Statement Let S denote (x,y,z) in R3 which satisfies the following inequalities: -2x+y+z <= 4 x-2y+z <= 1 2x+2y-z <= 5 x >=1 y >=2 z >= 3 Homework Equations How to find the dimension of the set S ? The Attempt at a Solution I have tried to transform the inequalities into matrix form...

I am trying to solve this inequality without using a factor table. The problem $$ \frac{x+4}{x-1} > 0 $$ The attempt at a solution As I can see ##x \neq 1##. I want to muliply both sides of the expression with x-1 to get rid of it, from the fraction. But before that, I have to consider two...

Homework Statement Homework Equations With the regards to posting such a incomplete equation, I will soon put in the updated one Thank you The Attempt at a Solution visual graph... didn't help

Homework Statement Hi Guys, This is the first exampe from Engel's problem solving book. After a long period of no math I am self studying. I do not know where my knowledge deficits lie, and was recommended this site for help. "E1. Starting with a point S (a, b) of the plane with 0 < b < a...

Hi! At university I have got a problem set with lots of inequalities. Unfortunately there are no explanations given how to do them. In Highschool we only did very easy inequalities. Therefore I am looking for a resource for inequalities. Especially for more difficult inequalities like $$ 1...

Homework Statement What values of X between 0 and 2 pie radians satisfy each of the following: 1. |sinX|<0.5 2. |cosX|>0.5 Homework Equations The Attempt at a Solution Well the values of X lie between 1. -0.5 < sinX <0.5 2. cosX< -0.5 and cosX>0.5 How do you find the...

Find where cosθ = -0.8660, where (0 degrees ≤ θ < 540 degrees) I am not sure how to solve this problem I did: cosθ + 0.8660 = 0 Then I am not sure what to do

Hi, can someone help me with the following question? I don't know how to approach the proof :confused: If a < b and c < d then ac < bd is true, supply a proof. Thanks!