Question

Lemma 2.17. Let k>0, and let f be a bounded function on the bounded interval [a,b]. Let P be a partition of [a,b]. Then (i) U(kf,P)=kU(f,P) and L(kf,P)=kL(f,P); and (ii) U(−f,P)=−L(f,P) and L(−f,P)=−U(f,P).

Answer 1

Lemma 2.17 states two properties of bounded functions on a bounded interval under a partition: (i) the upper and lower sums of k times the function f on the partition P are k times the upper and lower sums of f on the same partition P, and (ii) the upper sum of the negation of f on the partition P is the negative of the lower sum of f on the same partition P, and vice versa.

Step-by-step explanation:

Lemma 2.17 states two properties of bounded functions on a bounded interval under a partition:

1. For a positive constant k, the upper sum and lower sum of k times the function f on the partition P are k times the upper sum and lower sum of f on the same partition P.
2. For a negative constant k, the upper sum of the negation of f on the partition P is the negative of the lower sum of f on the same partition P, and vice versa.

These properties allow us to relate the upper and lower sums of a function f to the upper and lower sums of k times the function f, as well as the upper and lower sums of the negation of f. These properties are useful in the study of Riemann sums and the calculation of definite integrals.