Jay Cummings

This textbook is designed for students. Rather than the typical definition-theorem-proof-repeat style, this text includes much more commentary, motivation and explanation. The proofs are not terse, and aim for understanding over economy. Furthermore, dozens of proofs are preceded by "scratch work" or a proof sketch to give students a big-picture view and an explanation of how they would come up with it on their own. Examples often drive the narrative and challenge the intuition of the reader. The text also aims to make the ideas visible, and contains over 200 illustrations. The writing is relaxed and includes interesting historical notes, periodic attempts at humor, and occasional diversions into other interesting areas of mathematics. The text covers the real numbers, cardinality, sequences, series, the topology of the reals, continuity, differentiation, integration, and sequences and series of functions. Each chapter ends with exercises, and nearly all include some open questions. The first appendix contains a construction the reals, and the second is a collection of additional peculiar and pathological examples from analysis. The author believes most textbooks are extremely overpriced and endeavors to help change this. Hints and solutions to select exercises can be found at LongFormMath.com.

In the second issue of Into the Ruins, we continue to explore the emerging sub-genre of deindustrial science fiction. John Michael Greer debuts a new column on the sub-genre while five new and compelling stories weave future scenarios devoid of spaceships and interstellar space travel and instead focus on a future defined by natural limits, energy and resource depletion, industrial decline, climate change, and other consequences stemming from the reckless and shortsighted exploitation of our planet-and the ways humans will adapt, survive, live, die, and thrive within such a future. These stories show a wide variety of possible futures for humanity, from a distant civilization that cycles through the same ebb and flow of peace and warfare we find littered throughout human history, to a melancholic meditation on our fast-changing world set in 2020 that feels eerily familiar to today; from a love story set in a less energy-intensive time, to a haunting encampment at the edge of dry and dusty ruins; and on again to an adventurous and amusing attempt to deliver a key new manuscript on the herbal treatment of spinal meningitis to a distant library. These stories inspire a wide range of emotions, from meditative reflection on the predicament of our times to delight at unexpected adventure-all while offering readers fascinating tales utterly different than what science fiction normally provides.