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==The intermediate value theorem==
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==Limits, continuity, and convergence==
  
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As noted above, Gauss' 1816 proof of the fundamental theorem of algebra assumed as obvious, and hence did not prove, the intermediate value theorem. Bolzano was the first to offer a correct proof of the theorem, which he stated as follows:<ref>Jarnik</ref>
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Beginning perhaps with D'Alembert, it was an oft-repeated statement of 18th century mathematics that the calculus could be "based on limits." His own definition of limit is as follows:
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:If a function, continuous in a closed interval, assumes values of opposite signs at the endpoints of this interval, then this function equals zero at one inner point of the interval at least.
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:... the quantity to which the ratio z/u approaches more and more closely if we suppose z and u to be real and decreasing. Nothing is clearer than that.<ref>Dunham p. 72</ref>
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As has been noted elsewhere:<ref>Jarnik p. 36</ref>
 
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* the theorem seems ''intuitively'' plausible, for a continuous curve which passes partly under, partly above the x-axis, ''necessarily'' intersects the x-axis;
 
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* it was Bolzano's insight that the theorem needed to be proved ''as a consequence of the definition of continuity'' and he undertook to do in his paper of 1817.
 
  
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====Bolzano's proof====
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It is not surprising then that the arithmetization program culminated in the establishment of the concept of the limit and of those other fundamental concepts that were connected with it, namely, continuity and convergence.
  
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In his prefatory remarks, Bolzano discussed in detail previous proofs of the intermediate value theorem. Many of those proofs (alike with Gauss' 1799 proof of the fundamental theorem of algebra) depended "on a truth borrowed from geometry." Bolzano rejected all such proofs in totality and unequivocally:<ref>Russ p. 160</ref>
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====Limits====
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:It is an intolerable offense against correct method to derive truths of pure (or general) mathematics (i.e., arithmetic, algebra, analysis) from considerations which belong to a merely applied (or special) part, namely, geometry....  A strictly scientific proof, or the objective reason, of a truth which holds equally for all quantities, whether in space or not, cannot possibly lie in a truth which holds merely for quantities which are in space.
 
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Other proofs that Bolzano examined and rejected were based "on an incorrect concept of continuity":
 
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:No less objectionable is the proof which some have constructed from the concept of the continuity of a function with the inclusion of the concepts of time and motion.... No one will deny that the concepts of time and motion are just as foreign to general mathematics as the concept of space.
 
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Bolzano caped his prefatory remarks with the first mathematical achievement of his paper, namely, a formal definition of the continuity of a function of one real variable, which he stated as follows:<ref>Russ p. 162</ref>
 
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:If a function $f(x)$ varies according to the law of continuity for all values of $x$ inside or outside certain limits, then if $x$ is some such value, the difference $f(x + \omega) - f(x)$ can be made smaller than any given quantity provided $\omega$ can be taken as small as we please.
 
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Bolzano's proof of the main theorem proceeded as follows:
 
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* First, Bolzano introduced the (necessary and sufficient) condtition for the (pointwise) convergence of an infinite series, known today as the [[Cauchy test|Cauchy condition]] (on occasion the Bolzano-Cauchy condition), as follows:<ref>Russ p. 171</ref>
 
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:: If a series of quantities
 
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:::$F_1x$, $F_2x$, $F_3x$, . . . , $F_nx$, . . . , $F_{n+r}x$, . . .
 
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::has the property that the difference between its.$n$th term $F_nx$ and every later term $F_{n+r}x$, however far from the former, remains smaller than any given quantity if $n$ has been taken large enough, then there is always a certain constant quantity, and indeed only one, which the terms of this series approach, and to which they can come as close as desired if the series is continued far enough.
 
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:As noted previously, Bolzano demonstrated only the ''plausibility'' of the criterion, but did not provide a proof of its sufficiency.
 
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:Bolzano provided here also a proof of the fact that a sequence has at most one limit. The significance of this proof lies not in its achievement (since the proof is very easy) but in the fact that Bolzano may have been the first to realize the need for such a proof.<ref>Jarnik p. 36</ref>
 
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* Next, Bolzano used the [[Cauchy test|Cauchy condition]] in a proof of the following theorem:<ref>Russ p. 174</ref>
 
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::If a property $M$ does not belong to all values of a variable $x$, but does belong to all values which are less than a certain $u$, then there is always a quantity $U$ which is the greatest of those of which it can be asserted that all smaller $x$ have property $M$.
 
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:In effect, Bolzano here proved the greatest lower bound theorem. The number $U$ is in fact the greatest lower bound of those numbers which do not possess the property $M$.<ref>Jarnik p. 36</ref> The theorem proved is the original form of the [[Bolzano-Weierstrass theorem]] and is in fact the original statement of that theorem:<ref>Russ p.157</ref>
 
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::Every bounded infinite set has an accumulation point.
 
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:A complete proof of the greatest lower bound theorem, alike with the condition of convergence on which it depends, needed to await the building of the theory of real numbers. However, Bolzano here demonstrated the ''plausibility'' of the theorem.
 
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* Next, Bolzano stated and proved the intermediate value theorem, which he stated in a form that is now sometimes called Bolzano's theorem and that Bolzano himself believed to be "a more general truth" than the main theorem he set out to prove:<ref>Russ p. 177</ref>
 
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:If two functions of $x$, $f(x)$ and $g(x)$, vary according to the law of continuity either for all values $x$ or only for those which lie between $\alpha$ and $\beta$, and if $f(\alpha) < g(\alpha)$ and $f(\beta) > g(\beta)$, then there is always a certain value of $x$ between $\alpha$ and $\beta$ for which $f(x) = g(x)$
 
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* Finally, Bolzano proved the intermediate value theorem itself, which he stated in terms of the roots of a polynomial equation in one real variable, as follows:<ref>Russ p. 181</ref>
 
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:If a function of the form
 
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::$x^n + ax^{n-1} + bx^{n-2} + ... + px + q$
 
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:in which $n$ denotes a whole positive number, is positive for $x = \alpha$ and negative for $x = \beta$, then the equation
 
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::$x^n + ax^{n-1} + bx^{n-2} + ... + px + q = 0$
 
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:has at least one real root lying between $\alpha$ and $\beta$.
 
  
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====Cauchy's proof====
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Bolzano and Cauchy developed (independently) a concept of ''limit'' that had several advances over previous understandings:
 +
* it was free from the ideas of motion and velocity and did not depend on geometry
 +
* it did not retain the (unnecessary) restriction, that a variable could never surpass its limit
 +
Cauchy definition, in particular, stated only that the variable and its limit differed by less than any desired quantity, as follows:<ref>Grabiner (1981) p. 80</ref>
 +
:When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.
 +
Cauchy used this definition to define the infinitesimal as a dependent variable, thus freeing it from previous understandings of it as a very small fixed number:<ref>Boyer p. 540</ref>
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:One says that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge toward the limit zero.
  
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Quite independently of Bolzano, Cauchy formulated the intermediate value theorem in 1821 in the following, simpler form:<ref>[[Cauchy theorem]]</ref>
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Working with the notion of a sequence that "converges within itself," Bolzano and Cauchy sought to bind up the concepts ''limit'' and ''real number'', somewhat as follows:
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:If $f(x)$ is a continuous function of a real variable $x$ and $c$ is a number between $f(a)$ and $f(b)$, then there is a point $x$ in this interval such that $f(x) = c$.
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:If, for a given integer $p$ and for $n$ sufficiently large, S{n+p} differs from S{n} by less than any assigned magnitude $\varepsilon$, then S{n} also converges to the (external) real number $S$, the limit of the sequence.
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Indeed, some authors identify the theorem as Cauchy's (intermediate-value) theorem.
+
 
 +
Meray understood the error involved in the circular way that the concepts ''limit'' and ''real number'' were defined:<ref>Boyer p. 584</ref>
 +
* the limit (of a sequence) was defined to be a real number $S$
 +
* a real number was defined as a limit (of a sequence of rational numbers)
 +
Meray avoided references to convergence to an (external) real number $S$. Instead, he described convergence using only the rational numbers $n$, $p$, and $\varepsilon$, i.e. the Bolzano-Cauchy condition.
 +
 
 +
Weierstrass also understood the error involved in earlier ways of defining the concepts ''limit'' and ''irrational number'':<ref>Boyer p. 584</ref>
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* the definition of the former presupposed the notion of the latter
 +
* therefore, the the definition the latter must be independent of the former
 +
Roughly speaking, Weierstrass defined irrational numbers not as the limit of a series, but rather as the (convergent) sequence associated with the series. His definition of limit is as follows:
 +
:$lim{x→a}f(x) = L$ if and only if, for every $ε > 0$, there exists a $δ > 0$ so that, if $0 < |x - a| < δ$, then $|f(x) - L| < ε$.
  
 
==Notes==
 
==Notes==
 
<references/>
 
<references/>
  
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==Primary sources==
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==References==
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* Bolzano, Bernard (1817). ''Rein analytischer Beweis des Lehrsatzes, dass zwischen je zwey Werthen, die ein entgegengesetzes Resultat gewahren, wenigstens eine reelle Wurzel der Gleichung liege''. ["Purely analytic proof of the theorem that between any two values which give results of opposite sign there lies at least one real root of the equation"]. Prague.
 
  
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==References==
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* Bogomolny, A. "What Is Calculus?" from ''Interactive Mathematics Miscellany and Puzzles'' http://www.cut-the-knot.org/WhatIs/WhatIsCalculus.shtml#Alembert, Accessed 27 May 2014
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* Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem," ''Historia Mathematica'' 7 (1980), 156-185.
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* Dunham, W. (2008). ''The Calculus Gallery: Masterpieces from Newton to Lebesgue'', Princeton University Press.

Revision as of 21:18, 27 May 2014

Limits, continuity, and convergence

Beginning perhaps with D'Alembert, it was an oft-repeated statement of 18th century mathematics that the calculus could be "based on limits." His own definition of limit is as follows:

... the quantity to which the ratio z/u approaches more and more closely if we suppose z and u to be real and decreasing. Nothing is clearer than that.[1]

It is not surprising then that the arithmetization program culminated in the establishment of the concept of the limit and of those other fundamental concepts that were connected with it, namely, continuity and convergence.

Limits

Bolzano and Cauchy developed (independently) a concept of limit that had several advances over previous understandings:

  • it was free from the ideas of motion and velocity and did not depend on geometry
  • it did not retain the (unnecessary) restriction, that a variable could never surpass its limit

Cauchy definition, in particular, stated only that the variable and its limit differed by less than any desired quantity, as follows:[2]

When the successive values attributed to a variable approach indefinitely a fixed value so as to end by differing from it by as little as one wishes, this last is called the limit of all the others.

Cauchy used this definition to define the infinitesimal as a dependent variable, thus freeing it from previous understandings of it as a very small fixed number:[3]

One says that a variable quantity becomes infinitely small when its numerical value decreases indefinitely in such a way as to converge toward the limit zero.

Working with the notion of a sequence that "converges within itself," Bolzano and Cauchy sought to bind up the concepts limit and real number, somewhat as follows:

If, for a given integer $p$ and for $n$ sufficiently large, S{n+p} differs from S{n} by less than any assigned magnitude $\varepsilon$, then S{n} also converges to the (external) real number $S$, the limit of the sequence.

Meray understood the error involved in the circular way that the concepts limit and real number were defined:[4]

  • the limit (of a sequence) was defined to be a real number $S$
  • a real number was defined as a limit (of a sequence of rational numbers)

Meray avoided references to convergence to an (external) real number $S$. Instead, he described convergence using only the rational numbers $n$, $p$, and $\varepsilon$, i.e. the Bolzano-Cauchy condition.

Weierstrass also understood the error involved in earlier ways of defining the concepts limit and irrational number:[5]

  • the definition of the former presupposed the notion of the latter
  • therefore, the the definition the latter must be independent of the former

Roughly speaking, Weierstrass defined irrational numbers not as the limit of a series, but rather as the (convergent) sequence associated with the series. His definition of limit is as follows:

$lim{x→a}f(x) = L$ if and only if, for every $ε > 0$, there exists a $δ > 0$ so that, if $0 < |x - a| < δ$, then $|f(x) - L| < ε$.

Notes

  1. ↑ Dunham p. 72
  2. ↑ Grabiner (1981) p. 80
  3. ↑ Boyer p. 540
  4. ↑ Boyer p. 584
  5. ↑ Boyer p. 584

References

How to Cite This Entry:
Whayes43. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Whayes43&oldid=32223
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