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- Feb 29, 2012

- 340

Here's the question:

Let \( f \) be a bounded function on \( [a,b] \). Suppose there exist sequences \( (U_n) \) and \( (L_n) \) of upper and lower Darboux sums such that \( \lim (U_n - L_n) = 0 \). Show that \( f \) is integrable and that \( \int_a^b f = \lim U_n = \lim L_n \).

Here's my try:

By the hypothesis, exists \( M > 0 \) such that for all \(n > M, \varepsilon > 0 \) we have \( | U_n(f,P) - L_n(f,P) | < \varepsilon \), hence \( U_n(f,P) - L_n(f,P) < \varepsilon \) for some partition \( P \) of \( [a,b] \). It follows then that \( f \) is integrable, and by the limit properties we see that \( \lim(U_n - L_n) = \lim U_n - \lim L_n = 0 \implies \lim U_n = \lim L_n \).

My question is if that wouldn't imply already that \( \lim U_n = \int_a^b f \)? If not, I'm a bit lost. Would I have to show that for all \( n > M \) we have that \( L_n [f] \geq U_n [f] \), where \( L_n [f] \) and \( U_n[f] \) mean the lower and upper Darboux integrals respectively?

Also, that awkward moment when you type ( f ) without spaces and it becomes (f).