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--- math104-s21:s:antonthan [2021/05/07 13:35]
45.48.153.5 [Problem 17]
+++ math104-s21:s:antonthan [2022/01/11 18:30] (current)
24.253.46.239 ↷ Links adapted because of a move operation
@@ Line 169: / Line 169: @@
 === (a) ===
+Let partition $P_n = \{0, 1/n, 2/n, \ldots, n/n\}$.  Note that  $mesh(P_n) = 1/n$  and  $L(f, P_n) \leq R_n \leq U(f, P_n)$ .
+The idea now is to use Ross 32.7.  Let  $\epsilon > 0$ , and since  $f$  is integrable, there exists a  $\delta > 0$  that satisfies  $mesh(P) < \delta \implies U(f, P) - L(f, P) < \epsilon$  for all partitions  $P$ .  We choose  $n$  large enough such that  $mesh(P_n) < \delta$ , so that we get the inequalities:  $U(f, P_n) - \epsilon < R_n \leq U(f, P_n$  and  $L(f, P_n) \leq R_n < L(f, P_n) + \epsilon$ .  This gives us:
+ $U(f) \leq \sup_n(U(f, P_n)) = lim R_n = \inf_n(L(f, P_n)) \leq L(f)$
+Since we also have  $L(f) = U(f)$  by integrability,  $lim R_n = \int_0^1 f(x)dx$ , as desired.
 === (b) ===
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 ===== ඞ ඞ ඞ ඞ ඞ =====
-{{ :math104:s:amongus.mp4 |}}
+{{ math104-s21:s:amongus.mp4 |}}

Lecture Notes