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In many mathematical texts, I noticed that the quantifier "For all" and the phrase "For an arbitrary" are used interchangeably, and I have always thought of them as synonymous, as both indicate that the immediately following statement is true for any element in a particular set.

However, I recently found a textbook that says "For all" is more often used in a definition, and "For an arbitrary" should be placed at the beginning of proves. Are these terms not identical? If so, what are their differences?

The following text is from p.45 of Stephen Abbott, Understanding analysis (Vol. 11)

Learning to write a grammatically correct convergence proof goes hand in hand with a deep understanding of why the quantifiers appear in the order that they do. The definition begins with the phrase, “For all e> 0, there exists N ∈ N such that . . . ” Looking back at our first example, we see that our formal proof begins with, “Let e> 0 be an arbitrary positive number.” This is followed by a construction of N and then a demonstration that this choice of N has the desired property. This, in fact, is a basic outline for how every convergence proof should be presented. Template for a proof that (xn) → x :

  • Let e> 0 be arbitrary.”
  • Demonstrate a choice for N ∈ N. This step usually requires the most work, almost all of which is done prior to actually writing the formal proof.
  • Now, show that N actually works.
  • “Assume n ≥ N.”
  • With N well chosen, it should be possible to derive the inequality |xn − x| < .
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    $\begingroup$ When you are asked to prove a statement of the form “For all $x$, …”, you want to show that, for any $x$, whatever is in the ellipsis is true. To do this, we let $x$ be an arbitrary element, and then prove whatever it is we needed to prove. Since $x$ was arbitrary, we have proven it for all $x$. $\endgroup$ – twosigma Sep 16 at 6:21
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If a property P is true "for some arbitrary element" in a given set, then that will be true "for all" the elements in the set and vice-versa, so they are identical in the sense that they convey the same meaning. Now let's say someone told you that property P is true for some set A. well what will you do to prove him wrong, just find some element which doesn't satisfy that property and you are done. That's what is done in proving the property also but instead of finding an element which doesn't satisfy a property we chose an arbitrary element of a set and prove that the chosen arbitrary element satisfy the property and you are done. Since the element is chosen "arbitrarily" it will apply to every element. So it's just that the literature is different when you want to state the definition and when you want to formally prove something, but they are conveying the same meaning.

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