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.

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