User:Whayes43

In his paper of 1817, Bolzano undertakes to prove the following theorem about the roots of a polynomial equation in one real variable:

If a function of the form

$x^n + ax^{n-1} + bx^{n-2} + ... + px + q$ :in which $n$ denotes a whole positive number, is positive for $x = \alpha$ and negative for $x = \beta$, then the equation ::$x^n + ax^{n-1} + bx^{n-2} + ... + px + q = 0$ :has at least one real root lying between $\alpha$ and $\beta$. In his prefatory remarks, Bolzano discusses in detail previous proofs of the intermediate value theorem, many of which, including Gauss' first proof of 1799, depend "on a truth borrowed from geometry." Bolzano rejects all such proofs in totality and unequivocally:'"`UNIQ--ref-00000001-QINU`"' :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. In the process of proving the main theorem stated above, Bolzano achieves the following intermediate results:'"`UNIQ--ref-00000002-QINU`"' (a) states the formal definition of the continuity of a function of one real variable, as follows: :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. :Using notation he introduced in Section 14 of ''Binomische Lehrsatz'', Bolzano restates this as $f(x + \omega) = f(x) + \Omega$ (b) introduces the criterion for the (pointwise) convergence of an infinite series, as follows: : If a series of quantities ::$F_1x$, $F_2x$, $F_3x$, . . . , $F_nx$, . . . , $F_{n+r}x$, . . . :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. :As noted previously, Bolzano was here able to establish only the ''plausibility'' of the criterion. A proof of its sufficiency had to await the definition of the real number field. (c) states the [[Bolzano-Weierstrass theorem]] in its original form: :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 $M$ (d) states the intermediate value theorem in a form that is now sometimes called Bolzano's theorem, which Bolzano himself believed to be "a more general truth" than the main theorem he set out to prove: :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 $f(x) = g(x)$

Primary sources

• Bolzano, Bernard (1816). Der binomische Lehrsatz, und als Folqerunq aus ihm der polynomische, und die Reihen, die zur Berechnunq des Logarithmen und ExponentialqrBssen dienen, qenauer als bisher erwiesen. ["The binomial theorem, and as a consequence of it the polynomial theorem, and the series, which serve for the calculation of loqarithmic and exponential quantities, proved more strictly than before"]. Prague.
• 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.

References

• Russ, S. B. "A Translation of Bolzano's Paper on the Intermediate Value Theorem," Historia Mathematica 7 (1980), 156-185.