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.