I've got the function $f(x)=x^2-6x+8$ and the sequence of integrals   

 $I\_n=\int\_3^4 (x^2-6x+8)^n dx$   

$ n>0 $   

I have to prove that $I\_n$ is convergent. We are taught that a sequence converges provided it is monotone and bounded.  

My idea would be as follows: $f(x)$ is decreasing on $(3,4)$ and because n>0 we can say $f(x)^n$ is decreasing as well. To find the bounds I plug the values of the integral bounds, and since it is decreasing it would mean $f(4)<f(x)<f(3)$ and therefore $-1<x<0$. Is that right? And if so, is the answer complete?