I have a homework question which is:

> 
> If $f(x)$ is a bounded function in $[0,1]$ and $\sup
>  (f[\frac {1}{n},1])-\inf (f [\frac {1}{n},1])<\frac {1}{n}$ for
>  every natural $n>0$. Prove that $f(x)$ is Integrable in $[0,1]$.
> 
> 
> 

I have a feeling that this can be proven by showing somehow that $\inf(S\_n-s\_n)=0$ where $S\_n$ and $s\_n$ are the upper and lower limits of the split $P\_n$.

However I have not managed to prove this, perhaps I am attacking this problem from the wrong direction. 

Can someone help me out? 

Thanks :)