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Bolzano theorem

solakis

Active member
Dec 9, 2012
304
for the theorem (0.2) attached i have the following questions to ask:

1) In the case where f(xo)<0 ,the author justifies the contradiction he comes to,by writing that :

"every xε [xo,xo+δ) belongs to X"

How can we prove that?

2) In the case where f(xo)>0 ,the author justifies the contradiction he comes to,by assming that:

(xo-δ) is an upper bound for X.

How can we prove that?
 

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Opalg

MHB Oldtimer
Staff member
Feb 7, 2012
2,708
Re: bolzano theorem

for the theorem (0.2) attached i have the following questions to ask:

1) In the case where f(xo)<0 ,the author justifies the contradiction he comes to,by writing that :

"every xε [xo,xo+δ) belongs to X"

How can we prove that?

2) In the case where f(xo)>0 ,the author justifies the contradiction he comes to,by assming that:

(xo-δ) is an upper bound for X.

How can we prove that?
Both these things come from the definition of $X$, which says that if $x\in[a,b]$ satisfies $f(x)\leqslant0$ then $x\in X$. (But note that in both cases the author seems to be assuming that $\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$)
 

solakis

Active member
Dec 9, 2012
304
Re: bolzano theorem

Both these things come from the definition of $X$, which says that if $x\in[a,b]$ satisfies $f(x)\leqslant0$ then $x\in X$. (But note that in both cases the author seems to be assuming that $\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$)
yes,but how do you express mathematically the expression:

"$\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$"
 

solakis

Active member
Dec 9, 2012
304
Re: bolzano theorem

Both these things come from the definition of $X$, which says that if $x\in[a,b]$ satisfies $f(x)\leqslant0$ then $x\in X$. (But note that in both cases the author seems to be assuming that $\delta$ has been chosen small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$)

O.K suppose he chooses delta small enough so that the interval $(x_0-\delta,x_0+\delta)$ is contained in the interval $[a,b].$".

Then how does he get : SupX $\geq\delta+x_{o}> x_{o}$