Summary: Math 501. 1st Homework. Due Monday, September 10, 2007
Homework on "Chapter 1"
1. Let {xn}
n=1 be a sequence of real numbers. Let l R. Show that lim sup
n
xn = l if
and only if:
(i) for each > 0, there exists a positive integer n0 such that for each n n0
xn < l + .
(ii) for each > 0 and each positive integer n0 there exists a positive integer n n0
such that xn > l + .
2. Prove that for a sequence of subsets {An}
n=1 of the universal set , lim
n
An exists
if and only if for each , lim
n
I( An) exists.
Hint: Recall that I( An) is the indicator function of the set An.
3. Let A := {(x, y) R2

Source: Arcones, Miguel A. - Department of Mathematical Sciences, State University of New York at Binghamton

Collections: Mathematics