Proving Existence of Zero with Least Upper Bounds

[andilus]

Homework Statement

sup problem
if f is continuous on [a,b] with f(a)<0<f(b), show that there is a largest x in [a,b] with f(x)=0

Homework Equations

i think it can be done by least upper bounds, but i dun know wat is the exact prove.

The Attempt at a Solution

 

[rasmhop]

Start by showing that there is one x in [a,b] such that f(x) = 0. Then form the set
$$S = \{x \in [a,b] | f(x) = 0\}$$
You have already shown that S is non-empty and you know that it's bounded, so it must have a supremum. Let
$$x_0 = \sup\,S$$
Since $x_0$ is an upper bound for S, if we can show $x_0 \in S$ we have shown that it's the largest element in S. So all you need to do is show $x_0 \in S$. The easiest way to do this is to assume $x_0 \notin S$, i.e. $y_0 = f(x_0) \not= 0$. Now since f is continuous at $x_0$ we can find some $\delta > 0$ such that if $x \in (x_0-\delta,x_0 + \delta)$ then $f(x) \in (0,2y_0)$ (let $\epsilon = |y_0|$). Now $x_0 - \delta$ is an upper-bound for S, but less than $x_0$.

 

[andilus]

but how to prove x₀ is in [a,b]?

 

[andilus]

i seem to know...
if x₀<a, obviously wrong.
if x₀>b, we can find some x such that b-$$\delta$$<x<b satisfying that x is upper bound of the set.