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In measure theory, I saw that while proving some "equalities" - $``a=b"$ - (such as measure of any type of an interval is its length, ...), the argument goes as follows:

We prove that $a\leq b$ and $b\leq a$. One of these inequalities is mostly obvious, say $a\leq b$, and to prove second inequality, we prove that for every $\epsilon>0$, we have $b\leq a+\epsilon$.

I would like to know, instead of "measure theory theorems", are there elementary theorems in analysis, involving equality, whose proof runs in a way as described above?

Groups
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    Just note that, in analysis $a=b$ $\iff$ |a-b|<$\epsilon$, $\epsilon$>$0$ is arbitrary. – creative Aug 09 '14 at 08:49
  • May I ask why you are so interested in proofs of this particular structure? The idea behind it is very basic: if $x$ is smaller and greater than or equal to $y$, there simply is no other option than that $x=y$. – dreamer Aug 09 '14 at 08:50
  • @dreamer: I am going to teach Measure theory through Stein-Shakarachi, and in first chapter, I saw many proofs with same kind of argument. So, I can cover these theorems with a simple example. By the way, I was not too involved in analysis last 5 years, I don't know techniques in analysis. – Groups Aug 09 '14 at 08:52
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    In that case, students taking a Measure theory course really shouldn't have problems understanding that proof. If you really want an example, I suppose it is a way, if not an unnecessary one, to prove the uniqueness of a supremum in an real interval. – fixedp Aug 09 '14 at 09:02

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