# Elementary Theory of Metric Spaces

## A Course in Constructing Mathematical Proofs

• Robert B. Reisel
Book

Part of the Universitext book series (UTX)

1. Front Matter
Pages i-xi
2. Robert B. Reisel
Pages 1-13
3. Robert B. Reisel
Pages 14-33
4. Robert B. Reisel
Pages 34-51
5. Robert B. Reisel
Pages 52-58
6. Robert B. Reisel
Pages 59-68
7. Robert B. Reisel
Pages 69-74
8. Robert B. Reisel
Pages 75-83
9. Back Matter
Pages 85-121

### Introduction

Science students have to spend much of their time learning how to do laboratory work, even if they intend to become theoretical, rather than experimental, scientists. It is important that they understand how experiments are performed and what the results mean. In science the validity of ideas is checked by experiments. If a new idea does not work in the laboratory, it must be discarded. If it does work, it is accepted, at least tentatively. In science, therefore, laboratory experiments are the touchstones for the acceptance or rejection of results. Mathematics is different. This is not to say that experiments are not part of the subject. Numerical calculations and the examina­ tion of special and simplified cases are important in leading mathematicians to make conjectures, but the acceptance of a conjecture as a theorem only comes when a proof has been constructed. In other words, proofs are to mathematics as laboratory experiments are to science. Mathematics students must, therefore, learn to know what constitute valid proofs and how to construct them. How is this done? Like everything else, by doing. Mathematics students must try to prove results and then have their work criticized by experienced mathematicians. They must critically examine proofs, both correct and incorrect ones, and develop an appreciation of good style. They must, of course, start with easy proofs and build to more complicated ones.

### Keywords

Beweis /Aufgabensammlung Calc Compact space Connected space Metrischer Raum Spaces cluster compactness learning logic mathematical induction mathematics metric space theorem time

#### Authors and affiliations

• Robert B. Reisel
• 1
1. 1.Department of Mathematical SciencesLoyola University of ChicagoChicagoUSA

### Bibliographic information

• DOI https://doi.org/10.1007/978-1-4613-8188-4
• Copyright Information Springer-Verlag New York 1982
• Publisher Name Springer, New York, NY
• eBook Packages
• Print ISBN 978-0-387-90706-2
• Online ISBN 978-1-4613-8188-4
• Series Print ISSN 0172-5939
• Series Online ISSN 2191-6675