2 Mathematical Analysis Please provide complete and correct

2. Mathematical Analysis. Please provide complete and correct mathematical solution done on computer or by hand. Please, emphasis on correct solution. This is all the information provided. Thank you very much.

1. Give an example of a nested sequence of open intervals for which the intersection is (a) empty, b) a set with one member (c) an open interval (d) a closed interval (e) a semi-closed interval 2. Provide a proof for (a) and (b)

Solution

Consider the family An=(0,1n)An=(0,1n). We have An+1AnAn+1An for every nNnN, and the length of AnAnapproaches zero as nn approaches infinity, but n=1Ann=1An is empty. To see this, note that any element of the intersection would be greater than zero, yet less than 1n1n for every nNnN; by the Archimedian property of the real numbers, there is no such element.