# Muddiest Point 11/30

I thought that the muddiest point from last Thursday’s class was when we went over proving that $\int kf = k\int f$. The reason why I thought that this was the muddiest is because the proof required to cases. A case in which k was negative and a case that case was positive . This is often a detail that is easy to look over as it seems so trivial which is why it is important to go over it. To start the proof we assume that f is integrable and that $k>0$. Then we see that $U(kf,P) =\sum sup\{kf(x): x \in [x_{k-1},x_k] \}$. We already know from earlier this semester that $csup(A) = sup(cA)$ therefore we can say that $latex sup\{kf(x): x \in [x_{k-1},x_k] \} = ksup\{f(x): x \in [x_{k-1},x_k] \}$ and in turn $U(kf,P) = kU(f,P)$. we can use the same logic for the lower sums as well. Since $L(kf,P) = inf\{kf(x): x \in [x_{k-1},x_k] \}$ we can once again use our knowledge about infinums and supremums from earlier in the semester that $kinf(A) = inf(kA)$ to justify the statement that $inf\{kf(x): x \in [x_{k-1},x_k] \} = kinf\{f(x): x \in [x_{k-1},x_k] \}$ and $L(kf,P) = kL(f,P)$. Since the lower sums and Upper sums are unchanged by the location of the $k$ we can conclude that if $k$ is positive then $\int kf = k\int f$ and that there exists a partition $P_n$ such that $\lim_{n \to \infty} [U(kf, P_n) − L(kf, P_n)] = \lim_{n \to \infty} k[U(f,P_n) − L(f,P_n)] = 0$. Now we must consider the case that $k$ is negative. If k is negative $U(f,p)$ will be flipped when multiplied by k and in turn $L(f,P)$ will be flipped when multiplied by $k$. This distinction is important to make and is why the second case is needed. With the lower and upper sums flipped by the multiplication by $k$ we now have that $U(kf,P) = kL(f,P)$ and $kU(f,P) = L(kf,P)$. Therefore it is still the case that $\lim_{n \rightarrow \infty} [U(kf, P_n) − L(kf, P_n)] = \lim_{n \rightarrow \infty} k[U(f,P_n) − L(f,P_n)] = 0$. So we can conclude that $\int kf = k\int f$ regardless of the value of $k$.

#### One Response

1. Jeremy LeCrone says:

Thank you for the post. Good identification of a an issue that really is easy to overlook when you first consider this property.