Tricki

## Convert "every x" into a single arbitrary x

### Quick description

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

### Example 1

Suppose that you are asked to prove that whenever is a continuous function from to and is a sequence that converges to a limit , then converges to There is one step you can take before you start to think. More formally, the statement is "For every continuous function from to , and for every convergent sequence with limit , converges to ." The thought-free step is to write, "Let be a continuous function from to and let be a sequence that converges to a limit ." 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.