On this page, a proof of the equivalence of two definitions of Riemann integrals is given by the user Pedro using Apostol's Hint for Exercise 7.4, Mathematical Analysis. However, I still find this claim unclear (particularly, the first inequality):
$$S_1\leq U(P_\epsilon,f)<I+\frac \epsilon 2$$
To me, it seems that this is true only if f(x)>=0 on [a,b]. I'm sure I must be missing a subtle point here, so I would really appreciate it if anyone could clarify this for me.