# Real Analysis Solved Problems

See our librarian page for additional e Book ordering options. It is mainly intended for students studying the basic principles of analysis.However, given its organization, level, and selection of problems, it would also be an ideal choice for tutorial or problem-solving seminars, particularly those geared toward the Putnam exam. Each section of the book begins with relatively simple exercises, yet may also contain quite challenging problems.

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For instance, if one needs to prove a statement for all 0" title="\varepsilon , then there are an uncountable number of ‘s one needs to check, which may threaten measurability; but in many cases it is enough to only work with a countable sequence of s, such as the numbers for .

In a similar spirit, given a real parameter , this parameter initially ranges over uncountably many values, but in some cases one can get away with only working with a countable set of such values, such as the rationals.

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.

But they should be kept in mind as something to try if one starts having thoughts such as “Gee, it would be nice at this point if I could assume that is continuous / real-valued / simple / unsigned / etc. In the more quantitative areas of analysis and PDE, one sees a common variant of the above technique, namely the method of estimates.

Here, one needs to prove an estimate or inequality for all functions in a large, rough class (e.g. One can often then first prove this inequality in a much smaller (but still “dense”) class of “nice” functions, so that there is little difficulty justifying the various manipulations (e.g.

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.

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