Help with a continuity problem...

[DarkSun]

Hey there,
I am trying to solve the following continuity problem...
--------------------------------------------------------------------------------------
f,g:[a,b]->R
f,g are continous functions such that for each x in [a,b], f(x)<g(x).
Prove that exists c>0 such that for each x in [a,b] f(x)+c < g(x)
--------------------------------------------------------------------------------------
I have tried:
h(x) = g(x) - f(x), and so h(x) is continues and h(x) > 0.
Then probably some variation of the intermediate value theorem should be used, but I am not sure in what way?

Thanks.

 

[pka]

Note that the function \(\displaystyle h\) is also continuous on \(\displaystyle \left[ {a,b} \right]\).
Therefore, it has a minimum value: \(\displaystyle \left( {\exists q \in \left[ {a,b} \right]} \right)\left( {\forall x \in \left[ {a,b} \right]} \right)\left[ {0 < h(q) \leqslant h(x)} \right]\).
So let \(\displaystyle c = \frac{{h(q)}}{2}\).

 

[DarkSun]

Thanks