You are the mayor of el dorado. not all the streets are paved with gold only the interval [0,1] on main street- but you still have a serious security problem, and you ask heine-borel security llc to submit a proposal for keeping the street safe at night. knowing that the city coffers are full, they come up with the following pricey plan for meeting your requirements by using a countable infinity of guards:
a. guard 0 patrols the interval (-1, 1), where you may choose any value greater than 100 for the integer n. she is paid 200 dollars.
b. guard 1 patrols the interval (0.4,1.2) and is paid 100 dollars.
c. guard 2 patrols the interval (0.2,0.6) and is paid 90 dollars.
d. guard 3 patrols the interval (0.1,0.3) and is paid 81 dollars.
e. guard k patrols the interval (0.8/2k, 2.4/2k ) and is paid 100(0.9)k-1 dollars.
(a) calculate the total cost of hiring this infinite set of guards (sum a geometric series)
(b) show that the patrol regions of the guards form an "open cover" of the interval [0,1]
(c) according to the heine-borel theorem, this infinite cover has a finite subcover. explain clearly how to construct it. (hint: look at the proof of the heine-borel theorem)
(d) suppose that you want to protect only the open interval (0,1), which is not a compact subset of main street. in what very simple way can heine-borel security modify their proposal so that you are forced to hire infinitely many guards?