a repository of mathematical know-how

Convert "every x" into a single arbitrary x

Quick description

If you are trying to prove that some statement is true of every x of some kind, then a good way of organizing your thoughts is to begin by writing "Let x be" followed by a description of that kind of object. Then one can treat x as a single given object rather than feeling that one has to prove a statement for every x all at once. In other words, one replaces every x by an arbitrary x. Mathematically, this achieves nothing, but psychologically it helps.

Example 1

Suppose that you are asked to prove that whenever f is a continuous function from \mathbb{R} to \mathbb{R} and (x_n) is a sequence that converges to a limit x, then f(x_n) converges to f(x). There is one step you can take before you start to think. More formally, the statement is "For every continuous function f from \mathbb{R} to \mathbb{R}, and for every convergent sequence (x_n) with limit x, f(x_n) converges to f(x)." The thought-free step is to write, "Let f be a continuous function from \mathbb{R} to \mathbb{R} and let (x_n) be a sequence that converges to a limit x." Now we can concentrate on just this one (arbitrary) continuous function and this one (arbitrary) convergent sequence.

General discussion

This move is particularly useful in elementary real analysis – indeed, it should become an automatic reflex.