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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 $\mathbb R$: $ab$, or $a=b$. The $\leq$ relation for example means that $a

For example:

suppose that $a \leq b 0$. 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.

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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. – 2017-01-25
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    Sure but it's more a matter of practice and seeing it done than a singular answer. – 2017-01-25
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    I do not understand what you mean on the last sentence by "two are equal as well"? – 2017-01-25
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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. – 2017-01-26
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    Very clear proof. Thanks! – 2017-01-27