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# 9: Back to the Real Numbers

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• 9.1: Trigonometric Series
As we have seen, when they converge, power series are very well behaved and Fourier (trigonometric) series are not necessarily. The fact that trigonometric series were so interesting made them a lightning rod for mathematical study in the late nineteenth century.
• 9.2: Infinite Sets
All of our eﬀorts to build an uncountable set from a countable one have come to nothing. In fact many sets that at ﬁrst “feel” like they should be uncountable are in fact countable. This makes the uncountability of R all the more remarkable. However if we start with an uncountable set it is relatively easy to build others from it.
• 9.3: Cantor’s Theorem and Its Consequences
Once Cantor showed that there were two types of inﬁnity (countable and uncountable), the following question was natural, “Do all uncountable sets have the same cardinality?”

Thumbnail: Georg Cantor, German mathematician and philosopher of mixed Jewish-Danish-Russian heritage, the creator of set theory. Image used with permission (public domain).

## Contributors

• Eugene Boman (Pennsylvania State University) and Robert Rogers (SUNY Fredonia)