Preface

Natural numbers were created by God, everything else is the work of men.

- Isidor Kronecker

This volume contains notes on real analysis. Real analysis is considered one of the harder math courses, and many readers find it "dirty" compared to other fields of analysis, particularly its sibling, complex analysis. The "dirty" description likely stems from the fact that (1) results in real analysis are often ones that many readers are all very familiar with — ideas from elementary school through the typical calculus sequence, and (2) real analysis seems hell-bent on edge cases and counterexamples. As some readers will find, where complex analysis focuses on how beautiful and pleasing functions can be, real analysis focuses on how grotesque and pathological they can become.

That said, much of the modern world is built on real analysis. Reality isn't a pretty picture. It's complicated, grey, and messy. One might characterize real analysis as an accumulation of mathematical responses to this picture, and many of the simple operations and recipes we take for granted today — determining that one real number is greater than another, computing the slope of a curve, finding the area beneath a curve — exist only because real analysis has sanitized the kitchen.

For example, the ancient Greeks, while studying geometry for land surveying, stumbled upon the lengths ${\sqrt{2}}$ and ${\pi.}$ It's commonly held that the Pythagoreans, a religious cult of mathematicians to which we attribute Pythagoras's Theorem, deemed many of these numbers "evil," and attempted to hide these findings from the public. We can only conjecture as to whether this actually occurred, and if it did, why the Pythagoreans were so apprehensive towards their findings. A reasonable theory is that these numbers were simply problematic. They caused gaps in the Pythagorean view of numbers. And as is typical of religious cults, questions that cannot be answered should not be asked, and those that do ask them are labeled heretics. However, unlike notions of the divine, there's no way to hide mathematical results — if we don't discover them, someone else will eventually.

Indeed, numbers like ${\pi}$ and ${\sqrt{2}}$ kept appearing long after Greek antiquity, always accompanied by the question: What exactly is this number? In 1666, Isaac Newton answered: ${\pi}$ can be computed by summing an infinite series. But then the question was, what's an infinite series? Contrary to what one might think, neither Newton nor Leibniz had a clear understanding of a limit. This is not to underscore how monumental their works were to mathematics, but rather to highlight the fact that both mathematicians were constructing their ideas on extremely weak foundations. We can only imagine the fear and anxiety in these men's minds as they toiled.

Newton and Leibniz's works ushered in a new age of mathematics, one with ideas so radical and "dirty" that mathematicians of the old school — e.g., Kronecker — labeled them "the work of men," in contrast to the divinity embodied in number theory. Like the Pythagoreans, the apprehension from early modern mathematicians came from the difficulty of reasoning about such ideas. The infinite and the infinitesimals were a dark abyss, and there was no urgency to explore what lied therein.

That complacency came to an end in the early 1800s, when the French mathematician Joseph Fourier demonstrated the Fourier series — infinite sums of sines and cosines. Like the mathematicians of Newton's time, the 19th century mathematicians rejected Fourier's work. The difference, however, was that the non-mathematicians — the physicists and engineers — now sided with the underdog. The Fourier series solved numerous real-world problems that couldn't be solved with number theory. Standing from the upper echelons of their ivory towers, pure mathematicians saw Fourier's work spread like wildfire across applied mathematics, and realized that they could no longer ignore the dark world of infinites and infinitesimals.

Some time in the early 1820s, the French mathematician Augustin-Louis Cauchy spearheaded the effort towards clarifying the underlying foundations of Fourier's work. The notes in this volume focus heavily on Cauchy's work, and some of the results that followed.