Math 3A3 Solutions Fall 2012
Assignment 1 due 10:30am Friday September 21
q
1. (Exercise #3) If q rational satises q 2 = 12, then p = 2 is rational and satises
2
p2 = q4 = 3, so it su ces to prove there is no rational number p such that p2 = 3.
2
Suppose
Math 3A3 Solutions Fall 2012
Assignment 6
due Friday November 30
1. Let f : E ! R where E is a compact subset of R. Suppose rst that f is continuous.
By Theorem 29 on page 80 the function F : E ! R2 dened by F (x) = (x; f (x) is
continuous. Now the graph
Math 3A3 Solutions Fall 2012
Assignment 5 due Friday November 23
1. Let s0 > 0 and for n 2 N dene sn+1 = sn + s1 . Prove that the sequence fsn g1
n=0
n
diverges. To see this, we rst note that for any n 2 N, we have
(sn+1 )2 =
sn +
1
sn
2
= (sn )2 + 2 +
1
Math 3A3 Solutions Fall 2012
Assignment 2 due Friday October 5
1. Dene a cut to be a subset
the Lecture Notes:
of Q satisfying the three properties in (3.1) on page 9 of
(I)
6= ; and 6= Q;
(II) p 2
and q < p implies q 2 ;
(III) p 2
implies there is q 2 wi
Math 3A3 Solutions Fall 2012
Assignment 3 due Friday October 19
1. Let E be the set of all x 2 [0; 1] whose decimal expansion contains only 4 and 7
s
s.
(a) E is uncountable since it can be put into one-to-one correspondence with the
Cantor set C by mappi
Math 3A3 Solutions Fall 2012
Assignment 4 due Friday November 2
1. We must show that the function d dened on X n
inf f > 0 : K
d (K; L)
X n by the formula
L and L
satises the following properties for K; L; M 2 X n :
K g;
0
d (K; L) < 1;
d (K; L) > 0 i K 6