Proof involving continuity of two functions on the interval [a, b]?

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:

Quote:

[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 the problem:

Quote:

Given that

and

are both continuous on

, that

, and that

; then prove that

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:

Quote:

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