Chapter 10 Conclusion

Congratulations! You have come a long way since the definition of a limit point. The theorems that you proved on your journey are essential to many areas of mathematics including topology, complex analysis, functional analysis, real variables and measure theory. Perhaps as important as the results is the fact that you proved many of them on your own. Sadly, many an undergraduate has graduated with a degree in mathematics without the ability to either prove theorems on his or her own, or even understand the proof of a theorem as presented by another. Because of this, I know first-hand of graduate programs where a course equivalent to this course is taken for graduate credit because incoming students are unprepared to prove theorems. Even a student who graduates in mathematics without this skill should at least have a deep appreciation for this process that is so fundamental to the nature of the subject. Many undergraduate programs have avoided the issue of teaching students to prove theorems because of the difficulty of this daunting task. Applied programs often have minimal courses designed to train students in creating mathematics, rather they emphasize learning and applying mathematical results. While both have value, many of the best applied mathematicians are also pure mathematicians because not every problem is a direct application of a theorem. Sometimes the theorems must be modified or built to fit the application at hand.

Now let's talk about some of the important results that you have developed during this semester. I will speak loosely here without the precision to which you have been accustomed. Consider this a furtherance of your mathematical training. Many of the mathematicians you will encounter in the future will not be as precise with the language as we have been in this class. Just translate their work into nice precise mathematics just as you translated my rough proofs into precise mathematics when you let (made?) me present material on those rare days when you did not have mathematics to show off to the class. You can start practicing on what I have written below.

We started with limit points and convergence, two important underlying concepts in analysis and topology. And we played a bit with the study of open and closed sets. This is at the very heart of topology and analysis because we define continuity in terms of open sets. Therefore, if we change the definition of an open set, then we change the continuous functions. Go ahead, change “open interval” to “closed interval” in the definition of continuity and ask, “Which functions are continuous now?” The theorems that we proved that were topological in nature were Theorem 2.17 and Theorem 2.18 which show that the intersection of a nested sequence of closed intervals results in a point or a closed interval. Together these are referred to as the Nested Interval Theorem. In topology you will see generalizations of this — that the arbitrary union of open sets is open and the arbitrary intersection of closed sets is closed. In Theorem 2.19 you showed that no sequence could fill the closed interval, \([0,1]\text{.}\) This shows that the real line is not countable since a set is countable precisely when there is some sequence whose range is that set. Theorem 2.21 is the Bolzano-Wierstrauss Theorem and states that every infinite bounded set has a limit point. A consequence of this result that we used regularly was that every bounded sequence has a convergent subsequence.

We discussed four equivalent definitions of continuity, Definition 2.1, Definition 2.2, Definition 2.3 and Definition 2.11. The second will be generalized in a topological setting by writing that a function is continuous if and only if the inverse image of an open set under f is open. The third is the definition most often shown to calculus students. The fourth is the analyst's definition, that a function \(f\) is continuous if for every sequence converging to \(x\text{,}\) the sequence obtained by applying \(f\) to the original sequence converges to \(f(x)\text{.}\) You also proved that the sum of two continuous functions is continuous in Theorem 2.15. Along the way, we showed several properties of continuous functions. Together Theorem 4.32 and Theorem 4.33 showed that every continuous function on a closed interval has a maximum and a minimum value and attains those values. This is known as the Extreme Value Theorem. This theorem along with the Intermediate Value Theorem, Theorem 2.28, yielded that the range of a continuous function on a closed interval is either a point or a closed interval, Theorem 4.34.

Knowing that the maximum and minimum exist is not good enough. We must be able to find them, and for that we need derivatives. In calculus you talked a lot about tangent lines, but you probably did not accurately define a tangent line from a geometric point of view. If it was defined at all, it might have been defined by first defining the derivative in terms of limits and then defining the tangent line to \(f\) at \((x,f(x))\) to be the line with slope \(f'(x)\) passing through \((x,f(x))\text{.}\) Our approach was to offer a geometric definition of a tangent line in Definition 3.1 and then, if a function has a tangent line at a point, we offered a geometric definition of the derivative based on this tangent line. In total, you saw three equivalent definitions of the derivative, Definition 3.2, Definition 3.3, and Definition 3.4. You then proved that the derivative is unique in Theorem 3.8 and that every differentiable function is continuous in Theorem 3.15. As soon as you mastered derivatives in Calculus I, you started applying them to find the maxima and minima of differentiable functions and you always sought out points where the derivative was zero. In Theorem 3.16 we proved what you used in your calculus course: the derivative is zero at both local maxima and local minima.

The capstone theorem of the first semester was the fact that every continuous function is Riemann integrable, Theorem 4.24, a result we extended in a few ways. In fact, a function that is continuous on an interval except on a subset of that interval with measure zero is still integrable. This explains the introduction to measure theory. In the section on measure theory, we proved the Heine-Borel Theorem, Theorem 9.25, which states that any open cover of a closed interval has a finite subcover. We proved the Fundamental Theorem of Calculus, Theorem 4.40 and Theorem 4.42, the Mean Value Theorem for Integrals, Theorem 4.31, Rolle's Theorem, Lemma 4.35, and the Mean Value Theorem for Derivatives, Theorem 4.39. We showed that uniform continuity and continuity are equivalent on the interval, a result extending to any compact domain. We extended our notion of convergence to sequences of functions, defining pointwise convergence and uniform convergence. This led us to powerful theorems such as Theorem 5.16 and Theorem 6.4, which state that the uniform limit of continuous functions is continuous and that if the limit of a sequence of functions is uniform, then we may interchange the integral and the limit. We introduced uniform limits, Lipschitz functions, and series (Ratio Test, Theorem 5.5) to prepare us for a nice application of analysis, the existence and uniqueness of solutions to differential equations. While we showed this only for two very elementary differential equations, the process used illustrates the underlying concept for proving Picard's Existence Theorem.

Still, even with all the theorems we proved, we left out a few. Did you miss them? What about the continuity of the product and composition of continuous functions? What about the product, quotient, and composition (chain) rules for derivatives? What about the fact that we can factor a constant out of an integral? There are still many more such theorems to prove, but once you have proven a handful of theorems about continuity, derivatives, and integration, the rest fall in much the same way using the techniques that you learned this semester.

As I revisited these ideas in graduate school, the way all these theorems served as tools for other areas of mathematics, and the way they all were extended and generalized to spaces other than the real line, was part of the beauty of the subject. I hope that I have shared a part of the beauty of the subject with you and that it serves as a springboard to higher mathematics.