Let f a b rightarrow R be a bounded function Prove that the

Let f: [a, b] rightarrow R be a bounded function. Prove that the following conditions on f are equivalent: f is integrable on [a,b]; there exists a sequence (Q_n) of partitions of [a, b] such that lim(U(f, Q_n) - L(f, Q_n)) = 0 there exists a sequence (P_n) of partitions of [a,b] such that lim U(f, P_n) = lim L(f,P_n). Show also that if holds, then lim U(f,P_n) = lim L(f,P_n) is the integral of f on [a,b] Prove that the sequence (U(f, Q_n)) is bounded, and apply the Bolzano-Weierstrass Theorem. Prove and use statement (b).

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