When do I use "arbitrary" and/or "fixed" in a proof? In many proofs I see that some variable is "fixed" and/or "arbitrary". Sometimes I see only one of them and I miss a clear guideline for it. Could somebody point me to a reliable source (best a well-known standard book) which explains, when and how to use both in proofs?
EDIT: A little add-on to the question: Take a usual induction on natural numbers and assume that you are teaching it to students in their first semester. How do you explain "fixed" and "arbitrary" in this scenario?
 A: Both "arbitrary" and "fixed" are just shorthand for a universal quantifier. When I say something like "fix $\epsilon > 0$" it means I am about to prove a statement that is true for all $\epsilon > 0$ (and thereby prove that some function is continuous, for example) but I don't want to actually write out "for all $\epsilon > 0$" in front of every sentence I'm about to write. That's really all there is to it. 
A: When you encounter the term "arbitrary", it usually just means that a given statement is specified for any element from a given set of elments. For example, if I say, let $x$ be an arbitrary element of the interval $[0, 1]$ I just mean that $x$ can take on any value within that interval. 
The term fixed connotes a similar but more specific meaning. If I say let $x$ be a fixed element of the interval $[0, 1]$ I mean that, firstly, $x$ is an arbitrary element of $[0,1]$ but it's value is unchanging througout its usage.
The difference in meaning is subtle but sometimes important. If you start out with a "fixed" element you can't, for example, choose its value at some later point in the argument. On the other hand, if I prove that something holds for an arbitrary element of a given class, then it will hold for any particular element of that class which can often be very useful.
A: Rather then arbitrary and fixed, lets look at something very similar where the difference is more pronounced.  There is a very important distinction between uniform and fixed.  A perfect example is convergence of functions.
Let $f_n$ be a sequence of functions of a set $X$.  For each point $x\in X$, let $f$ be defined by $f(x)=\lim_{n\rightarrow \infty} f_n(x)$.  That is, $f$ is the pointwise limit, and it is the limit for fixed $x$.
However, uniform convergence is much stronger, and requires that the limit is small for all $x$ simultaneously.
Very often things come up which are easy to prove for fixed $x$, but very difficult when we need uniformity in $x$.
Hope that helps,
A: *

*The proof of a universally-quantified (“for each $x$”) statement might begin with one of these:

*

*Let $x$ be an arbitrary (i.e., any) real number greater than $7.$

*Consider any (i.e., an arbitrary) real number $x$ greater than $7.$
The idea is that this representative value is fixed only while applying the remaining steps, and that for each iteration of the proof's flow of logic a value is chosen for $x.$


*The proof is eventually concluded by invoking Universal Introduction to convert that opening sentence to

*

*For each $x{>}7,\ldots,$
which finally refers generally to every element of $(7,\infty).$


*How to interpret “let... suppose” in mathematics?
An overanalysis of the various synonyms of “for each"
