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 :)