In Adams book we see a definition:
if it is just one $I$ that for every partition $P$ of $[a,b]$ we have: $L(f,P)$$<=$$I$$<=$$U(f,P)$ we say that f is integrable in $[a,b]$.but we see that every one prove this by saying that $L(f,P_{n})=U(f,P_{n})$ ($n\rightarrow\infty$).
how this two are equal.(as Adams book say after we prove that the limits are equal we should prove that for every partition of $[a,b]$ I is between $L$ and $U$. so one part of proof is incomplete)