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This booklet by way of Jakob Nielsen (1890-1959) and Werner Fenchel (1905-1988) has had
a lengthy and intricate background. In 1938-39, Nielsen gave a sequence of lectures on
discontinuous teams of motions within the non-euclidean airplane, and this led him - in the course of
World conflict II - to write down the 1st chapters of the booklet (in German). while Fenchel,
who needed to get away from Denmark to Sweden as a result of the German career,
returned in 1945, Nielsen initiated a collaboration with him on what grew to become recognized
as the Fenchel-Nielsen manuscript. at the moment they have been either on the Technical
University in Copenhagen. the 1st draft of the Fenchel-Nielsen manuscript (now
in English) used to be entire in 1948 and it was once deliberate to be released within the Princeton
Mathematical sequence. besides the fact that, as a result of the fast improvement of the topic, they felt
that gigantic alterations needed to be made ahead of booklet.
When Nielsen moved to Copenhagen collage in 1951 (where he stayed until eventually
1955), he used to be a lot concerned with the overseas association UNESCO, and the
further writing of the manuscript was once left to Fenchel. The records of Fenchel now
deposited and catalogued on the division of arithmetic at Copenhagen Univer-
sity include unique manuscripts: a partial manuscript (manuscript zero) in Ger-
man containing Chapters I-II (

I -15), and a whole manuscript (manuscript I) in
English containing Chapters I-V (

1-27). The documents additionally comprise a part of a corre-
spondence (first in German yet later in Danish) among Nielsen and Fenchel, the place
Nielsen makes unique reviews to Fenchel's writings of Chapters III-V. Fenchel,
who succeeded N. E. Nf/Jrlund at Copenhagen college in 1956 (and stayed there
until 1974), used to be a great deal concerned with an intensive revision of the curriculum in al-
gebra and geometry, and targeted his examine within the thought of convexity, heading
the overseas Colloquium on Convexity in Copenhagen 1965. for nearly twenty years
he additionally placed a lot attempt into his task as editor of the newly begun magazine Mathematica
Scandinavica. a lot to his dissatisfaction, this task left him little time to complete the
Fenchel-Nielsen undertaking the way in which he desired to.
After his retirement from the collage, Fenchel - assisted via Christian Sieben-
eicher from Bielefeld and Mrs. Obershelp who typed the manuscript - came across time to
finish the booklet undemanding Geometry in Hyperbolic house, which used to be released by way of
Walter de Gruyter in 1989 presently after his loss of life. at the same time, and with an identical
collaborators, he supervised a typewritten model of the manuscript (manuscript 2) on
discontinuous teams, elimination a number of the imprecise issues that have been within the unique
manuscript. Fenchel informed me that he meditated removal elements of the introductory
Chapter I within the manuscript, considering the fact that this is able to be coated by means of the booklet pointed out above;
but to make the Fenchel-Nielsen booklet self-contained he eventually selected to not do
so. He did choose to pass over
27, entitled Thefundamental team.

As editor, i began in 1990, with the consent of the criminal heirs of Fenchel and
Nielsen, to provide a TEX-version from the newly typewritten model (manuscript 2).
I am thankful to Dita Andersen and Lise Fuldby-Olsen in my division for hav-
ing performed a superb activity of typing this manuscript in AMS- TEX. i've got additionally had
much aid from my colleague J0rn B0rling Olsson (himself a pupil of Kate Fenchel
at Aarhus college) with the evidence interpreting of the TEX-manuscript (manuscript three)
against manuscript 2 in addition to with a basic dialogue of the difference to the fashion
of TEX. In such a lot respects we made up our minds to persist with Fenchel's intentions. notwithstanding, turning
the typewritten version of the manuscript into TEX helped us to make sure that the notation,
and the spelling of yes key-words, will be uniform through the booklet. additionally,
we have indicated the start and finish of an explanation within the traditional sort of TEX.
With this TEX -manuscript I approached Walter de Gruyter in Berlin in 1992, and
to my nice aid and delight they agreed to post the manuscript of their sequence
Studies in arithmetic. i'm such a lot thankful for this optimistic and quickly response. One
particular challenge with the ebook became out to be the replica of the various
figures that are an essential component of the presentation. Christian Siebeneicher had at
first agreed to convey those in ultimate digital shape, yet through 1997 it turned transparent that he
would no longer have the capacity to locate the time to take action. although, the writer provided an answer
whereby I may still carry special drawings of the figures (Fenchel didn't go away such
for Chapters IV and V), after which they'd arrange the construction of the figures in
electronic shape. i'm very thankful to Marcin Adamski, Warsaw, Poland, for his nice
collaboration about the genuine construction of the figures.
My colleague Bent Fuglede, who has personaHy recognized either authors, has kindly
written a quick biography of the 2 of them and their mathematical achievements,
and which additionally locations the Fenchel-Nielsen manuscript in its right standpoint. In
this connection i want to thank The Royal Danish Academy of Sciences and
Letters for permitting us to incorporate during this e-book reproductions of photos of the 2
authors that are within the ownership of the Academy.
Since the manuscript makes use of a few certain symbols, a listing of notation with brief
explanations and connection with the particular definition within the publication has been integrated. additionally,
a complete index has been additional. In either circumstances, all references are to sections,
not pages.
We thought of including an entire record of references, yet determined opposed to it as a result of
the overwhelming variety of examine papers during this sector. as a substitute, a miles shorter
list of monographs and different accomplished debts suitable to the topic has been
My ultimate and such a lot honest thank you visit Dr. Manfred Karbe from Walter de Gruyter
for his commitment and perseverance in bringing this ebook into lifestyles.

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Additional resources for A First Course in Mathematical Analysis

Example text

4. Use the Principle of Mathematical Induction to prove that, for n ¼ 1, 2, . . : (a) 12 þ 22 þ 32 þ Á Á Á þ n2 ¼ nðnþ1Þ6ð2nþ1Þ; qffiffiffiffiffiffiffiffi qffiffiffiffiffiffiffiffi 5=4 1Á3Á5Á ... Áð2nÀ1Þ 3=4 (b) 4nþ1 2Á4Á6Á ... Áð2nÞ 2nþ1. À2 5. Apply Bernoulli’s Inequality, first with x ¼ 2n and then with x ¼ ð3nÞ to prove that 2 2 1 3n 1 þ ; for n ¼ 1; 2; . . : 1þ 3n À 2 n 6. By applying the Arithmetic Mean–Geometric Mean Inequality to the n þ 1 positive numbers 1, 1 À 1n , 1 À 1n , 1 À 1n , . . , 1 À 1n, prove that nþ1 À Án  1 1 À 1n 1 À nþ1 ; for n ¼ 1; 2; .

First, we have to prove that a is an upper bound of E. To do this, we prove that, if x > a, then x 2 = E (this is equivalent to proving, that, if x 2 E, then x a). We begin by representing x as a non-terminating decimal x ¼ x0 Á x1x2 . .. Since x > a, there is an integer n such that a < x0 Á x1 x2 . . xn : Hence x0 Á x1 x2 . . xn ! a0 Á a1 a2 . . an þ 1 ; 10n and so, by our choice of an, x ¼ x0 Á x1 x2 . . xn is an upper bound of E. Since x > x0 Á x1x2 . . xn, we have that x 2 = E, as required.

Hence it is sufficient to prove the following statement P(n) for each natural number n: P(n): For any positive real numbers ai with a1a2 . . an ¼ 1, then a1 þ a2 þ Á Á Á þ an ! n. First, the statement P(1) is obviously true. Next, we assume that P(k) holds for some k ! 1, and prove that P(k þ 1) is then true. Now, if all the terms a1, a2, . , akþ1 are equal to 1, the result P(k þ 1) certainly holds. Otherwise, at least two of the terms differ from 1, say a1 and a2, such that a1 > 1 and a2 < 1.

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