JL 1

Ulisse Dini, 1845-1918 , Italy

Dini’s theorem (not in book)

Let (fn : R → R)n∈N a sequence of continuous functions pointwisely converging to a and such that ∀n ∈ N, ∀x ∈ [a, b], fn+1(x) ≥ fn(x). Then (fn : R → R)n∈N converges uniformly.

One interesting fact about this mathematician: Beside being a mathematician, Dini reached the highest office in university administration when he became rector of the , he was elected to the national Italian parliament in 1880 as a representative from Pisa. He was the chair of “infinitesimal analysis”. Another interesting fact about this mathematician: The implicit function theorem is known in Italy as the Dini’s theorem. How many stars you give to your mathematicians: ERIC COOKE 2

Thomas Joannes Stieltjes, 1865- 1894 The Netherlands

Definition of the Riemann-Stieltjes sum (35.24, p.320) Let f be bounded on [a, b], and let P = {a = t0 < t1 < . . . < tn = b} , a partition of [a, b].A Riemann-Stieltjes sum of f associated with P and F is a sum of the form n n X + −  X − +  f (tk ) F(tk ) − F(tk ) + f (xk ) F(tk ) − F(tk−1) . k=0 k=1

where xk is in (tk−1, tk ) for k = 1, 2,..., n.

One interesting fact about this mathematician: Stieltjes never graduated college and in fact failed out twice. It was his achievements in that earned him an honorary degree. How many stars you give to your mathematicians: I gave this mathematician four stars, mainly because he died so young and only worked in the field for less than ten years. SYD FREDERICK 3

Michel Rolle, 1652-1719

Rolle’s Theorem (29.2, p.233) Let f be a continuous function on [a, b] that is differentiable on (a, b) and satisfies f (a) = f (b). There exists [at least one] x in (a, b) such that f 0(x) = 0.

One interesting fact about this mathematician: Educated himself in Mathematics, no formal training.

How many stars you give to your mathematicians: 5 out of 5, because his theorem is very fundamental and helps to prove the Mean Value Theorem. He also was one of the first mathematicians to publish Gaussian elimination in Europe. JOHN GORDOS 4

Julius Wilhelm Richard Dedekind, 1831-1916

Dedekind Cuts (§6, p.30) Dedekind Cuts are a way to define the real numbers from the rational numbers. A Dedekind cut A is a subset of Q satisfying these properties: 1. A is neither ∅ nor Q; 2. If r is in A, s is in Q and s < r, then s is in A; 3. A contains no largest rational. The set of all possible Dedekind cuts can be used as the definition of R.

One interesting fact about this mathematician: Dedekind was the last student of Gauss. How many stars you give to your mathematicians: Building the reals like this is mindblowing to think about, more so because Dedekind acknowledged he had weaknesses in advanced mathematics after receiving his doctorate. From here, he spent two years studying to compensate. I sympathize but my weakness exists on a foundational level. KJERSTI JACOBSON 5

Georg Cantor, 1845-1918 Germany

Cantor set (Example 5, p.89) In 1883, he introduced the concept of the Cantor set. The Cantor set is simply a subset of the interval [0, 1], but the set has some very interesting properties: for instance, the set is compact, uncountable, and contains no intervals. The most common modern construction of a Cantor set is the Cantor ternary set, which is built by removing the middle thirds of a line segment.

One interesting fact about this mathematician: Cantor believed that Francis Bacon wrote Shakespeare’s plays. He studied intensely Elizabethan literature to try to prove his theory. In 1896-97 he published pamphlets on the subject.

How many stars you give to your mathematicians: XINXIN JIANG 6

Brook Taylor, 1685-1731

Taylor (31.2,p.250) Let f be a function defined on some open interval containing c. If f possesses derivatives of all orders at c, then the Taylor series for f about c is ∞ X f (k)(c) (x − c)k . k! k=0

One interesting fact about this mathematician: As a mathematician, he was the only Englishman after Sir Isaac Newton and Roger Cotes capable of holding his own with the Bernoullis; but a great part of the effect of his demonstrations was lost through his failure to express his ideas fully and clearly. How many stars you give to your mathematicians: I give him 4. Though it is very important for a mathematician to focus on mathematical research, a good grasp of communications skills is also vital. And unfortunately, he is not good at expressing himself despite his brilliant thinking process. SIYUAN LIN 7

Jean Gaston Darboux, 1842-1917 France

Intermediate Value Theorem for Derivatives (29.8,p.236) Let f :(a, b) → R be a differentiable function. If a < x1 < x2 < b, and if c lies between 0 0 0 f (x1) and f (x2), then there exists (at least one) x in (x1, x2) such that f (x) = c. Upper Darboux sums (p.270) Given a f : R → R, given a partition of [a, b], P = {a = t0 < t1 < . . . < tn = b}, the upper Darboux sum U(f , P) of f with P is the sum

n ! X U(f , P) = sup f (x) (tk − tk−1) . k=1 x∈[tk−1,tk ]

One interesting fact about this mathematician: In 1902, he was elected to the Royal Society; in 1916, he received the from the Society. How many stars you give to your mathematicians: His theorems seem really complicated since we learn it at the very end of this book, so I guess he must be really brilliant. And he must be a great professor, because he taught many highly reputed European mathematicians, for example, Émile Borel, Élie Cartan, Gheorghe ¸Ti¸teicaand Stanisław Zaremba. So he deserves five stars. SAMUEL LOOS 8

Georg Friedrich Bernhard Riemann, 1826-1866 German

Riemann integral (p.270) Given L(f ) (resp. U(f )) the lower (resp. upper) Darboux integral of f over [a, b], we say that f is (Riemann) integrable on [a, b] provided that L(f ) = U(f ). In this case, we write Z b f = L(f ) = U(f ). a

One interesting fact about this mathematician: The base of Einstein’s Theory of Relativity was set up in 1854 when Riemann gave his first lectures on the of space.

How many stars you give to your mathematicians: I would give Riemann 4 stars. His contribution to numerous areas in mathematics is immense. He also had a lot of influence with the development of prime numbers. CARLY MEYER 9

Karl Weierstrass, 1815-1897 German

Bolzano-Weierstrass Theorem (11.5, p 72) Every bounded sequence has a convergent subsequence. Weierstrass M-test (25.7, p 205) Let (gk : → )k∈ be a sequence of functions and (Mk )k∈ a sequence of real numbers such that R R N P P N (1) for all x ∈ R, |gk (x)| ≤ Mk and (2) MK < ∞, then gK converges uniformly. Weierstrass’s Approximation Theorem (27.5, p 220) Every continuous function on a closed interval [a,b] can be uniformly approximated by polynomials on [a.b].

One interesting fact about this mathematician: Along with teaching mathematics, he taught physics, gymnastics, geography, history, German, calligraphy and botanics at the Lyceum Hosianum in Braunsberg, Poland. A second interesting fact about this mathematician: Weierstrass has a lunar crater named after him. How many stars you give to your mathematicians: I give Weierstrass 5 out of 5 stars because he played a significant role in a lot of the content that we have learned this semester. Without the Bolzano-Weierstrass theorem, a lot of our proofs would fall apart. The Weierstrass M-test is also quite a strong theorem–without knowing the pointwise convergence of a sequence of functions, we still have the ability to conclude if a sequence of functions converges uniformly. SAMUEL MORTELLARO 10

Bernard Bolzano, 1781-1848 Prague, Kingdom of Bohemia

Bolzano-Weierstrass theorem (11.5, p.72) Every bounded sequence has a convergent subsequence.

One interesting fact about this mathematician: Because he argued adamantly that war was a human and economic waste, he was exiled to the county side and not allowed to publish in mainstream journals. For this reason, most of his works only became well known posthumously. How many stars you give to your mathematicians: Four, I would give a random moderately famous and important mathematician a three. I gave Bolzano a four because he went beyond just the field of mathematics, and applied mathematical thinking to philosophy. He developed a rigorous theory of science and became a formative influence on analytic philosophy; a philosophical movement which I think deserves credit for removing the nonsense and ambiguity from continental philosophy (please note: that is a lot of nonsense) and has survived to this day. KATHERINE PAINE 11

Augustin-Louis Cauchy, 1789-1857 France

Cauchy sequence (10.8, p.62)

A sequence (sn)n∈N of real numbers is called a Cauchy sequence if and only if

∀ε > 0, ∃N, ∀m > N, ∀n > N, |sn − sm| < ε.

One interesting fact about this mathematician: There exist sixteen concepts and theorems named after him, more than any other mathematician.

How many stars you give to your mathematicians: 5 stars because we consantly see his definition/theorems show up throughout the class. The Cauchy sequence concept has showed up for sequences, series, uniform continuity, . LAWRENCE PELO 12

Emile Borel, 1871-1956 France

Heine-Borel Theorem (13.12, p.90) k A subset E of R is compact if and only if it is closed and bounded.

One interesting fact about this mathematician: He served for 12 years in the French National Assembly, and was a member of the French Resistance during World War II. A second interesting fact about this mathematician: Borel worked on the Infinite Monkey Theorem, which states that a monkey hitting keys at random on a typewriter keyboard for an infinite amount of time will almost surely type a given text, such as the complete works of William Shakespeare.

How many stars you give to your mathematicians: 5 stars, for his founding work in . MIKAEL SPETH 13

Eduard Heine, 1821-1881 Germany

Heine-Borel Theorem (13.12, p.90) k A subset E of R is compact if and only if it is closed and bounded. Heine-Cantor Theorem (19.2, p.143) If f is a continuous function on [a, b], then f is uniformly continuous on [a, b].

One interesting fact about this mathematician: His advisor for his studies was another famous mathematician Peter Dirichlet

How many stars you give to your mathematicians: He came up with some interesting theorems/results but I did not find much more information about him. JENNA TSEDENSODNOM 14

Niels Henrik Abel, 1802-1829 Norway

From notebook of Niels Abel Abel’s theorem (26.6, p.212) P n Let f (x) = anx be a power series with finite positive radius of convergence R. If the series converges at x = R, then f is continuous at x = R. If the series converges at x = −R, then f is continuous at x = −R.

One interesting fact about this mathematician: At the age of 16, Abel gave a proof of the binomial theorem valid for all numbers, extending Euler’s result which had only held for rationals.

How many stars you give to your mathematicians: CHANG WANG 15

Isaac Newton, 1643-1727 England

Newton’s Method (31.8, p.259) Newton’s method for finding an approximate solution to f (x) = 0 is to begin with a reasonable initial guess x0 and then compute

f (xn−1) xn = xn−1 − 0 , for n ≥ 1. f (xn−1)

Often the sequence (xn)n∈N converges rapidly to a solution of f (x) = 0. One interesting fact about this mathematician: Newton believed in magic. In addition to his more respectable scientific pursuits, Newton was a student of alchemy and the occult. How many stars you give to your mathematicians: 5 stars. He was not only a great mathematician but also a great physical scientist and astronomer. He was an all-round talent who laid the foundation for physics, mathematics, and engineering. LIXIN WANG 16

Joseph-Louis Lagrange, 1736-1813 France

Taylor’s theorem with Lagrange remainder (31.3,p.250) (n) Let f : R → R be defined on (a, b). Suppose the n-th derivative f exists on (a, b), we denote Rn(x) the remainder of the Taylor series of f about c. Then for each x 6= c in (a, b) there is some y between c and y such that:

f (n)(y) R (x) = (x − c)n . n n!

One interesting fact about this mathematician: He has said that “if I had been rich, I probably would not have devoted myself to mathematics”. How many stars you give to your mathematicians: Lagrange made significant contributions to the fields of analysis, , and both classical and celestial mechanics. We will meet many theorem and methods of him in our math courses. SHUXIAN YANG 17

Jacques Hadamard, 1865-1963 France

Cauchy-Hadamard Theorem (23.1,p.188) P n For the power series anx , let

1 1 β = lim sup |a | n and R = . n β

[If β = 0 we set R = +∞, and if β = +∞ we set R = 0.] Then (i) The power series converges for |x| < R; (ii) The power series diverges for |x| > R.

One interesting fact about this mathematician: He married his childhood sweetheart.

How many stars you give to your mathematicians: YUNQUN YI 18

Guillaume de l’Hôpital, 1661-1704 , France

l’Hôpital’s Rule (30.2,p.241) + − Let s signify a, a , a , ∞ or −∞ where a ∈ R, and suppose f and g are differentiable functions for 0 which the following limit exists: lim f (x) = L. (Note that this hypothesis includes some implicit x→s g0(x) assumptions: f and g must be defined and differentiable “near” s and g0(x) must be nonzero “near” s.) f (x) If limx→s f (x) = limx→s g(x) = 0 or if limx→s |g(x)| = +∞, then limx→s g(x) = L.

One interesting fact about this mathematician: L’Hôpital abandoned a military career due to poor eyesight. In 1691 he met young Johann Bernoulli, who was visiting France and agreed to supplement his Paris talks on infinitesimal with private lectures to l’Hôpital at his estate at Oucques. How many stars you give to your mathematicians: I will give 5 stars to him. Because he is very smart and his method about the calculus is very useful. On the other hand, he works very hard even though he has poor eyesight.