Looks good!
I was wondering if somebody could check my proof for the following problem. I'm failry certain its correct (atleast correct in the general direction of the proof), I just wanted to make sure. Firstly, the proof makes use of the following theorems / properties:
Now heres the problem:
[1]
If is continuous on (i.e. for all ) and is continuous on then is continuous on .
[2]
If is continuous on , and , then for some
Now heres my proof. I know this problem will probably seem trivial and super easy to allot of people, but I just wanted to make sure I'm doing it correctly:
Given that and are both continuous on , that , and that ; then prove that for some
Showing that:
For some is equivelant showing that:
For some
So, lets let:
By the given information and property/theorem number [1], we know that is continuous on
Now, since:
then:
And since:
then:
So we have:
Equivilantly...
Since we also know that is continuous on , then by property/theorem [2] we are garunteed that:
For some
Thus:
For some
Therefore we've shown we must have:
For some