Intermediate value theorem proof

have a continuous function f:[0,1] -> [0,1]

For some
For some

Using the Intermediate Value Theorem, show that there exists some x* such that f(x*)=x*

I have attempted the problem:

Make . Then we are looking for the case when . So if f(0)=0 or f(1)=1, then we are done. So we assume neither case is true. Then we have that because f must take on a value in the interval [0,1], and the same for . These two statements imply that . So then by the intermediate value theorem, we can say that for some which means that for some

As I am pretty rusty on my proof skills I was wondering if somebody could check this for me.

Re: Intermediate value theorem proof

Using the Intermediate Value Theorem, show that there exists some x* such that f(x*)=x*

I have attempted the problem:

Make . Then we are looking for the case when . So if f(0)=0 or f(1)=1, then we are done. So we assume neither case is true. Then we have that because f must take on a value in the interval [0,1], and the same for . These two statements imply that . So then by the intermediate value theorem, we can say that for some which means that for some

As I am pretty rusty on my proof skills I was wondering if somebody could check this for me.

Thanks

The function is continuous on an interval containing both and , with and , hence by theIntermediate value theorem there exists a between and such that and so