 
Summary: 18.014ESG Problem Set 4
Pramod N. Achar
Fall 1999
Monday
1. Prove the linearity property for integrals of bounded functions, as follows:
(a) Let A R be nonempty and bounded above, and let c R be some
fixed number. Show that if we define C = {cx  x A}, then C is
nonempty and bounded above, and supC = c sup A.
(b) Let A and B be two nonempty, bounded above subsets of R. Show
that if we define C = {x+y  x A and y B}, then C is nonempty
and bounded above, and sup C = sup A + sup B.
(c) Using the previous two parts and the linearity property for integrals
of step functions, show that if f1 and f2 are bounded functions on
[a, b] and c1, c2 R, then
b
a
(c1f1 + c2f2) = c1
b
a
f1 + c2
