Epsilontics

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An epsilon or ε environment around the number  a ,
drawn on the number line

The epsilontics is a term from the analysis . It is used to formulate terms such as limit value or continuity with mathematical precision. Using epsilontics, terms such as “infinitely small” or “less than any given positive number” are specified. The term epsilontics is sometimes used in a slightly disparaging manner when the routine character of evidence is to be emphasized. The amount of deviation from the limit value is usually denoted by the Greek letter epsilon . To z. As the convergence of a real consequence to the limit to prove it shows that for every little number with a number so there is that for each applies: . ${\displaystyle (\varepsilon )}$ ${\displaystyle \!\ f_{n}}$ ${\displaystyle \!\ f}$ ${\displaystyle \varepsilon >0}$ ${\displaystyle \varepsilon \in \mathbb {R} }$ ${\displaystyle \!\ n_{0}\in \mathbb {N} }$ ${\displaystyle \!\ n>n_{0}}$ ${\displaystyle |f_{n}-f|<\varepsilon }$

Or in the two common quantifier notations:

1	${\displaystyle \bigwedge _{\varepsilon >0}}$	${\displaystyle \bigvee _{n_{0}}}$	${\displaystyle \bigwedge _{n>n_{0}}}$	${\displaystyle \left|f_{n}-f\right|<\varepsilon }$
2	${\displaystyle \forall \varepsilon >0}$	${\displaystyle \exists n_{0}}$	${\displaystyle \forall n>n_{0}:}$	${\displaystyle \left|f_{n}-f\right|<\varepsilon }$
read as:	For every epsilon greater than zero	if there is an n -zero for which applies that	for all n greater than n -zero:	The amount of f n minus f is less than epsilon.

Since proofs for the convergence of a sequence in epsilontics often contain the sentence “be ”, there are jokes like the variant “be ” (which seems absurd to a mathematician) . ${\displaystyle \varepsilon >0}$ ${\displaystyle \varepsilon <0}$

Epsilontics goes back to Karl Weierstrass , who first introduced the epsilon environments to define the limit value. This definition only requires that variables be in a certain range and no longer says that variables are moving towards a limit value. If one had previously argued intuitively with ideas of movement, now the notation of epsilontics put the concept of limit values ​​on a stable mathematical foundation, which enabled an exact problem definition and provability .

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1. ↑ Epsilontics . In: Guido Walz (Ed.): Lexicon of Mathematics . 1st edition. Spectrum Academic Publishing House, Mannheim / Heidelberg 2000, ISBN 3-8274-0439-8 .
2. ↑ Harro Heuser: Textbook of Analysis. Part 2. B. G. Teubner, Stuttgart 1990, ISBN 3-519-42222-0 , p. 696 f.