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Before I read the proof of Bolzano's theorem from my Calculus book, I've tried to prove it myself. I will use the following lemma and the least upper bound axiom.

[Lemma: Sign-preserving property of continuous functions.] Let $f$ be continuous at $c$ and suppose that $f(c) \neq 0$. Then there is an interval $(c - \delta , c + \delta )$ about $c$ in which $f$ has the same sign as $f(c)$.

[Bolzano's theorem] Let $f$ be continuous at each point of a closed interval $[a, b]$ and assume that $f(a)$ and $f(b)$ have opposite signs. Then there is at least one $c$ in the open interval $(a, b)$ such that $f(c) = 0$.

Proof:

(I assume that the sign of $f(a)$ is negative, and consequently $f(b)$ has a positive sign.)

1) Let $\mathbf{N}$ be the set of all points of $[a,b]$ where $f \le 0$. Then $a \in \mathbf{N}$ and $b \notin \mathbf{N}$. Since $b$ is the maximum element of the interval, $b$ is an upper bound for $\mathbf{N}$. Hence $\mathbf{N}$ has a supremum. And the supremum is strictly less than $b$ and strictly greater that $a$ (we can use the lemma to show this).

2) Let $s = sup\mathbf{N}$ and we know that $a < s < b$. Then since $f$ is continuous for every point in $[a,b]$, $f$ must be defined for every such point, and $f(s)$ has a value.

3) And I say that $f(s) = 0$. For if not, either $f(s) < 0$ or $f(s) > 0$.

4) If $f(s) > 0$ then according to the lemma there is a positive number $\delta$ such that for any value in $(s - \delta, s]$, $f > 0$. ($s - \delta$ must be greater than $a$ since $f(a) < 0$). Then none of these values from the interval is in $\mathbf{N}$. Hence for any $x \in \mathbf{N}, x \le s - \delta$. We have found an upper bound which is less than the supremum. It is a contradiction.

5) In a similar way we show that if $f(s) < 0$ than there must be numbers in $[a,b]$ bigger than $s$ for which $f < 0$, so they must be in $\mathbf{N}$.

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  • $\begingroup$ Don't use $\Bbb N$, is the usual symbol for set of natural mumbers!. You can use $N$. $\endgroup$ Feb 4, 2014 at 11:11
  • $\begingroup$ Small change: "Since $b$ is the maximum element of the interval... $\endgroup$ Feb 4, 2014 at 11:13
  • $\begingroup$ Looks good to me. To be a little nitpicky: "Then since $f$ is defined for every point in $[a,b]$, $f(s)$ has a value." $\endgroup$
    – Dejan Govc
    Feb 4, 2014 at 11:38
  • $\begingroup$ I've read the proof from my textbook. It's the same. Differs only in small details. $\endgroup$
    – mosceo
    Feb 4, 2014 at 12:24

1 Answer 1

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(Some people use $\mathbf{N}$ for natural numbers instead of the $\mathbb{N}$ symbol mentioned in the comments, too. $N$ is better, and something like $A$ would be even more better because $N$ often stands for a number $\in \mathbb{N}$, not a set of numbers, but that's a question of taste. Also, $\sup A$ (with \sup) instead of $sup A$.)

I think the proof is correct. For what it's worth, instead of assuming $f(a) <0 $ and $f(b) >0$, with a slight modification it suffices to assume $f(a)$ and $f(b)$ have opposite signs:

Let $a' \in \{a, b\}$ such that $f(a') = \min\{f(a), f(b)\}$, and likewise $b'$ such that $f(b') = \max\{f(a), f(b)\}$.

Now clearly $f(a') < 0$ and $f(b') > 0$, and one can proceed as you've done, but working with $a'$ instead of $a$ and so on.

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