*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 .So remember to exercise some care with the epsilon of room trick when some quantities are infinite. If one has to prove something about an unbounded (or infinite measure) set, consider proving it for bounded (or finite measure) sets first if this looks easier.*

Specific topics covered in this volume include the following: basic properties of real numbers, continued fractions, monotonic sequences, limits of sequences, Stolz's theorem, summation of series, tests for convergence, double series, arrangement of series, Cauchy product, and infinite products. Nowak&type=R Would be an ideal choice for tutorial or problem-solving seminars. presentation of material is designed to help student comprehension and to encourage them to ask their own questions and to start research …

Also available from the AMS are https://org/exam-desk-review-request? a really useful book for practice in mathematical analysis.

Ordering on the AMS Bookstore is limited to individuals for personal use only.

Libraries and resellers, please contact [email protected] assistance. This book is the first volume of a series of books of problems in mathematical analysis.

The collection of problems in the book is also intended to help teachers who wish to incorporate the problems into lectures. The book covers three topics: real numbers, sequences, and series, and is divided into two parts: exercises and/or problems, and solutions.

Specific topics covered in this volume include the following: basic properties of real numbers, continued fractions, monotonic sequences, limits of sequences, Stolz's theorem, summation of series, tests for convergence, double series, arrangement of series, Cauchy product, and infinite products.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 ).For instance, an error term such as is certainly OK, or even more complicated expressions such as if one has the ability to choose as small as one wishes, and then after is chosen, one can then also set as small as one wishes (in a manner that can depend on ).One caveat: for finite , and any 0" title="\varepsilon and , but this statement is not true when is equal to (or ). Decompose or approximate a rough or general object by a smoother or simpler one.Uncountable unions are not well-behaved in measure theory; for instance, an uncountable union of null sets need not be a null set (or even a measurable set).(On the other hand, the uncountable union of sets remains open; this can often be important to know.) However, in many cases one can replace an uncountable union by a countable one.(See also the Tricki for some general mathematical problem solving tips.Once this page matures somewhat, I might migrate it to the Tricki.) Note: the tricks occur here in no particular order, reflecting the stream-of-consciousness way in which they were arrived at. If one has to show that two numerical quantities X and Y are equal, try proving that and separately. If one has to show that , try proving that for any 0" title="\varepsilon .Similarly, the Cauchy-Schwarz inequality can flip a upper bound on to a lower bound on , provided that one has a lower bound on .Holder’s inequality can also be used in a similar fashion. Uncountable unions can sometimes be replaced by countable or finite unions.

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