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In class, I mentioned that when solving the type of homework problems encountered in a graduate real analysis course, there are really only about a dozen or so basic tricks and techniques that are used over and over again.

This trick works best when the objects being reflected are contained in some sort of “bounded”, “finite measure”, or “absolutely integrable” container, so that one avoids having the dangerous situation of having to subtract infinite quantities from each other.

The triangle inequality can be used to flip an upper bound on to a lower bound on , provided of course one has a lower bound on .

Sometimes one needs a lower bound for some quantity, but only has techniques that give upper bounds.

In some cases, though, one can “reflect” an upper bound into a lower bound (or vice versa) by replacing a set contained in some space with its complement , or a function with its negation (or perhaps subtracting from some dominating function to obtain ).

In the latter case, one may need an epsilon of room (trick 2), and some sort of limiting analysis may be needed to deal with the errors in the approximation (it is not always enough to just “pass to the limit”, as one has to justify that the desirable properties of the approximating object are preserved in the limit).

: one should not do this blindly, as one might then be loading on a bunch of distracting but ultimately useless hypotheses that end up being a lot less help than one might hope.

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