My Math Forum  

Go Back   My Math Forum > College Math Forum > Calculus

Calculus Calculus Math Forum

Thanks Tree4Thanks
  • 3 Post By Country Boy
  • 1 Post By agent1594
LinkBack Thread Tools Display Modes
July 12th, 2016, 12:53 PM   #1
Joined: Oct 2014

Posts: 13
Thanks: 2

Lightbulb Direct proof for this inequality theorem

Theorem:$\displaystyle a\leqslant b+\epsilon, \forall \epsilon > 0 \Rightarrow a\leqslant b $ where $\displaystyle a,b\in \mathbb{R}$

I have seen proof by contradiction for this. But I tried to give a direct proof as follows;

Let $\displaystyle a,b\in \mathbb{R} \wedge \epsilon > 0$
$\displaystyle a\leqslant b+\epsilon \Rightarrow a-b\leqslant \epsilon$
That is $\displaystyle \epsilon_{min}=a-b$
But since $\displaystyle \epsilon > 0$,
$\displaystyle \epsilon_{min} > 0 \geqslant 0$
$\displaystyle \therefore a-b\geqslant 0 \Rightarrow a\geqslant b$

Here I obtained a complete opposite of the theorem.
Could someone please point me where I have gone wrong?
agent1594 is offline  
July 12th, 2016, 01:05 PM   #2
Math Team
Joined: Jan 2015
From: Alabama

Posts: 3,264
Thanks: 902

You cannot write "$\displaystyle \epsilon_{min}= b- a$" unless you first show that there is a minimum value for $\displaystyle \epsilon$ (and there isn't).
Thanks from topsquark, agent1594 and manus

Last edited by Country Boy; July 12th, 2016 at 01:10 PM.
Country Boy is offline  
July 12th, 2016, 10:53 PM   #3
Joined: Oct 2014

Posts: 13
Thanks: 2

Is that the condition "$\displaystyle \epsilon > 0$" which keeps us away from concluding that $\displaystyle \epsilon_{min}=a-b$ ?
And what if we didn't have that condition and the statement was as follows?
$\displaystyle a\leqslant b+\epsilon, \forall \epsilon > a-b \Rightarrow a\leqslant b $ where $\displaystyle a,b\in \mathbb{R}$
Is the above statement legit?
Thanks from manus
agent1594 is offline  

  My Math Forum > College Math Forum > Calculus

direct, direct proof, epsilon, inequality, inequality theorem, proof, theorem

Search tags for this page
Click on a term to search for related topics.
Thread Tools
Display Modes

Similar Threads
Thread Thread Starter Forum Replies Last Post
triangle inequality theorem ehh Abstract Algebra 4 November 2nd, 2014 11:30 AM
Discrete math direct proof question mgk501 Real Analysis 2 March 21st, 2013 12:23 PM
Prove it, Without Cauchy's inequality, Liouville's theorem bigli Complex Analysis 5 February 1st, 2012 05:51 AM
Direct Proof? jrklx250s Real Analysis 3 December 3rd, 2011 03:58 AM
proof by using greens theorem or stokes theorem george gill Calculus 5 May 14th, 2011 02:13 PM

Copyright © 2019 My Math Forum. All rights reserved.