I have been through a couple of proofs by now. I wonder why mathematicians need to prove something to be equal, by proving it can be higher or equal and less or equal. What is the point? They make use of epsilon all the time for that purpose. I would be grateful for answer.
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There is something known as the axiom of trichotomy. It states that one of the following relations holds on $\mathbb R$: $a<b,a>b$, or $a=b$. The $\leq$ relation for example means that $a<b$ or $a=b$. The result is similar for $\geq$. Therefore if you can show that some number is simultaneously less $\leq$ and $\geq$ another, the two must actually be equal. The $\epsilon$ is common in real analysis, the point is that if $a \leq b <\epsilon$ for all $\epsilon$, the two are equal as well.
For example:
suppose that $a \leq b <a+\epsilon$ for all $\epsilon>0$. Suppose to the contrary that $a \neq b$. Let $\epsilon =(b-a)/2$, and notice that $b>a+\epsilon$, a contradiction. Thus, $a=b$.
Another typical thing:
Let $\emptyset \neq A \subseteq \mathbb R$. Suppose that $b\geq a$ for all $a \in A$. Also, suppose that for all $\epsilon>0$, there exists some $a_{\epsilon} \in A$ so that $a_{\epsilon}>b-\epsilon$. Then $b$ is the least upper bound.
Andres Mejia
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Thank you for your answer. Could you develop a little bit the epsilon explanation in the last sentence? Why is b less than epsilon? Sorry for not writing with math terminology. – Pedro Gomes Jan 25 '17 at 16:03
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Sure but it's more a matter of practice and seeing it done than a singular answer. – Andres Mejia Jan 25 '17 at 19:55
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I do not understand what you mean on the last sentence by "two are equal as well"? – Pedro Gomes Jan 25 '17 at 21:50
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I think that it would be beneficial to review the least upper bound property, but you can see my edits for potential clarification. – Andres Mejia Jan 26 '17 at 04:28
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Very clear proof. Thanks! – Pedro Gomes Jan 27 '17 at 16:25