If I have a function $f$ that is integrable on [a,b], how can I show that it is also integrable on some interval [a,c], for some point c $\in$ (a,b)?
My attempt:
For all $\epsilon > 0$, there exists a partition P such that $U(f,P)-L(f,P)<\epsilon$. There exists a partition P' such that for all $\epsilon > 0$, $U(f,P')-L(f,P')<\epsilon$. The point $c \in [a,b]$, so define P' to be all the points of P up to and including c, then it becomes apparent that this is true.
Is this correct and rigorous enough? I feel that it's not much more than intuition but I'm not sure how to construct a more detailed proof.
Thanks for your time.