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I have been assigned this problem for homework:

Show that, if $a < b + \epsilon$ for every $\epsilon \gt 0$, then $a\le b$.

I have tried to go about this using Induction, but I don't know what the base case would be. It is obvious to me in my mind, but I don't know how to put it into mathematical terms on paper. any hints?

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4  
Presumably $a$ and $b$ are arbitrary real numbers. If so, induction is not a suitable tool. –  André Nicolas Jan 25 '13 at 21:00
    
(Standard) induction only works for statements that can be phrased as "for every $n$, P(n)" where $n$ is a natural number. Since $\epsilon$ is allowed to be any real number >0, induction won't work. –  icurays1 Jan 25 '13 at 21:02
    
What strategy would you recommend. –  Sam Jan 25 '13 at 21:08
    
@Sam - take a look at the answers below. –  icurays1 Jan 25 '13 at 21:10

4 Answers 4

Hint: Use contradiction. If $a>b$, then show $a$ is not less than $b+\frac{a-b}{2}$. This contradiction implies $a\leq b$.

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I'm a snob, so I like using contrapositive instead of contradiction whenever possible. The contrapositive would go like this: "if $a>b$, then there exists an $\epsilon>0$ such that $a\geq b+\epsilon$." Finding this epsilon should be easy with a number line.

Edit: the original poster figured it out in the comments, so here's the full argument: if $a>b$, then certainly $a-b>0$. Hence set $\epsilon=a-b$; then, since $b+\epsilon=b+a-b=a$, we certainly have $a\leq b+\epsilon=a$ (since $a=a$).

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we could choose an actual number for ϵ or a representation using a and b? –  Sam Jan 25 '13 at 21:12
    
A representation using $a$ and $b$, exactly - you can't pick say "$\epsilon=0.1$" because if $a$ and $b$ are closer, that $\epsilon$ won't work. –  icurays1 Jan 25 '13 at 21:14
    
Should we let ϵ be equal to $a-b$? that way a would be greater than equal to $b+(a-b)$ therefore $a \geq a$ –  Sam Jan 25 '13 at 21:16
    
Exactly - which is true, since $a=a$. I'll write the whole argument in the answer. –  icurays1 Jan 25 '13 at 21:17
    
thank you so much ! I knew using induction was leading us down the wrong path –  Sam Jan 25 '13 at 21:18

I'm assuming this is from a first proof-based course, so apologies if this comes off as condescending.

Induction is harder to use in this case for precisely the reason you stated: what is the base case? We don't have any equivalent of some smallest nonzero number 1 with which to start like we do in $\mathbb{N}$. Maybe we could start with 0.0001? No, 0.00001 is smaller. And 0.000001 is smaller still. This sort of idea is known as the Well-Ordering Principle and is maybe not immediately clear, but it's worth thinking about. In particular, $\mathbb{R}$ (the real numbers) is not well-ordered, since there is no smallest element in $(0,1)$.

But in statements involving inequalities it's often a good idea to try to do things by contradiction. In that case, we'd assume $a > b$ and try to show that this is impossible. Well, if $a > b$ then $a - b > 0$. But we are given that $a < b + \epsilon$ for any $\epsilon > 0$. So what value should we choose for $\epsilon$ to find our contradiction?

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assume a > b. Then a-b > 0. Then let ϵ = a - b. Then a < b + a-b = a , a contradiction so a <= b.

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