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