(adsbygoogle = window.adsbygoogle || []).push({}); 1. The problem statement, all variables and given/known data

Let [tex]x,y\in R[/tex] such that [tex]x\leq y+\epsilon[/tex] for all [tex]\epsilon>0[/tex]. Prove that [tex]x\leq y[/tex].

2. Relevant equations

None other than what's given in the question.

3. The attempt at a solution

It's obvious to me that it's true because I can pick some numbers [tex]x>y[/tex] so that the original inequality is not true for all [tex]\epsilon[/tex]. I've tried proof by contradiction, but it doesn't leave to any contradictions. For example, assuming that [tex]x>y[/tex], then I've come up with the following things:

- [tex]y<x\leq y+\epsilon[/tex] or [tex]\epsilon>0[/tex] (this is okay from the assumptions in the problem)
- [tex]x+\epsilon>y+\epsilon\geq x[/tex] since [tex]x>y[/tex] and [tex]x\leq y+\epsilon[/tex] (this is okay since [tex]\epsilon>0[/tex])
- [tex]y-\epsilon<x-\epsilon\leq y[/tex] or [tex]y-\epsilon<y[/tex] (again okay from the assumptions of the problem).

I've also tried to write it in terms of quantifiers, negate it and prove the contrapositive, but I'm not at all sure what I'm doing since I've never formally covered it. I've come up with the following as an equivalent statement for the problem.

[tex]\forall x,y,\epsilon\in R, \epsilon>0 : x\leq y+\epsilon \Rightarrow x\leq y[/tex]

My attempt to negate it (which is probably wrong) is

[tex]\exists x,y,\epsilon\in R, \epsilon>0 : x\geq y+\epsilon\Rightarrow x>y[/tex].

Can anybody give me any help and/or let me know if I'm doing anything right? Thanks in advance!

**Physics Forums | Science Articles, Homework Help, Discussion**

Dismiss Notice

Join Physics Forums Today!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

The friendliest, high quality science and math community on the planet! Everyone who loves science is here!

# Homework Help: Classical Analysis I, difficult inequality

**Physics Forums | Science Articles, Homework Help, Discussion**