140 - Fall 2007

DUE TUESDAY OCT 2:
Page 5: 1,3,6, 8 (a,b), 11.  Hint on (b): use (a).


DUE THURSDAY, OCT 4:
HW#0: Prove the square root of 7 is not rational.
HW#1: Prove that for any prime p, the square root of p is not rational.

Page 18, 3.1 (a,b)

DUE TUESDAY, OCT 9:
Page 18, 3.3, 3.4, 3.5, 3.6

HW#2: Definition: A set S is inductive if
whenever a natural number n is in S, then n+1 is also in S.

a) Give 3 examples of inductive sets.
b) Revisit the statement of Problem 1.11 (page 5) in light of this definition.  
What is Problem 1.11 "really" about?

DUE THURSDAY, OCT 11:

4.1 (a,b,e,g,m)
4.2 (a,b,e,g,m)
4.3 (a,b,e,g,m)
4.4 (a,b,e,g,m) NO PROOF NECESSARY FOR ANY OF THE ABOVE PROBLEMS.

4.5,
4.8,
4.12 (this problem is now due Tuesday, Oct 16)

DUE TUESDAY, OCT 16:

Prove the Approximation Theorem for Sup's:
Let S be a bounded, non-empty set.  Let U = sup{S}.  Then prove
for every e > 0 there exists an s in S such that

U-e < s =< U.

Hint:  Consider two cases: U in S, U not in S.

4.12,

DUE TUESDAY, OCT 16:

7.1, 
7.2 
7.3 (a,b,c,d,s,t)
7.4

DUE THURSDAY, OCT 18:

8.1(a,c)
8.2(a,b)
8.3
8.4

DUE TUESDAY, OCT 23:


8.9(a)
8.10
8.5
9.4,
9.5,
9.6,

HW#4: Prove that for all n in the natural numbers and all positive numbers b,
(1+b)^n >= 1+nb.  Hint: what kind of proof is this likely to be?


DUE THURSDAY, OCT 25:
9.8 (a,d)
9.9 (a,c)
9.10 (a)
9.11 (a)
9.12 (a,b)



MIDTERM FRIDAY OCT 26 - bring a small bluebook to class


DUE TUESDAY, OCT 30:

10.1, 
10.4 (discuss thm 10.2 ONLY, not 10.11)
10.5

DUE THURSDAY, NOV 1:
10.7,
10.8, 


10.9
10.11


EXTRA DRILL PROBLEMS ON lim inf AND lim sup:

For each of the following sequences, determine the lim inf and the lim sup, by
writing the first few u's and v's:

a) {1,2,1,3,1,4,...}
b) {1, 2, 1/2, 3, 1/3, ... }
c) {1, -1, 2, -2, 3, ... }
d) { 1, -1/2, 1/3, -1/4, 1/5, ... }


DUE TUESDAY, NOV 6:

10.6 (a)
10.6 (b) <-- explain this
11.1
11.2 (a)
11.6
11.7

DUE THURSDAY, NOV 8:

HW#5: (more on Bolzano-Weierstrass:).  Let {s_n} be the sequence 
{1/2, 2/3, 3/4, ...}  Find a suitable A and B such that A <= s_n <= B
for n = 1, 2, ...  .  (i.e. find bounds for the sequence.)

With those bounds, compute I_1, I_2, I_3, and I_4 with the property 
from the proof of the theorem:  

{n: s_n in I_k is infinite} for k = 1, 2, 3, 4.

Now compute a_1, a_2, a_3, a_4.   What is lim {a_k}?

Can you construct a subsequence of the original sequence that converges to that limit?


DUE TUESDAY NOV 13:

11.2,
11.5
11.10

HW#6: Prove: L is a subsequential limit (limit point) of the sequence {s_n: n = 1,2,...} 
if and only if for every epsilon > 0 and all M,

there exists an n > M  such that |s_n - L| < epsilon.

DUE THURSDAY NOV 15:

hw#7: a) Let f(1/n) = n, for n = 1, 2, ...  (domain of f is 
{1, 1/2, 1/3, 1/4, ...}.  Prove f is continuous!

b)  Suppose in addition, that f(0) = c is defined (so that 0 is now in the domain of f).  Prove that
f is not continuous at 0.

17.1, 17.9(a)
17.10 (a,b)
17.12 (a,b)
17.13


DUE TUESDAY NOV 20:

17.2(a,b)
17.3
17.5
17.13 (again!)
17.12 (a,b) (again!)

MIDTERM#2: WEDNESDAY, NOV 21

DUE TUESDAY, NOV 27:
18.1, 18.2, 18.3, 18.5, 18.6, 18.9 (by "root", we mean a place where the function takes the value 0).

DUE TUESDAY, DEC 4:
19.1 (a,b,c,d)
19.4 (a)
19.5 (f)
19.6 (b)
19.7 (a)

Optional - try 19.6 (a)

20.1, 20.11(a,b), 20.13 (a,b), 20.14

DUE TUESDAY, DEC 4 ALSO (was due Thursday, Nov 29 but I forgot to post this):

19.2 (a,b,c)

DUE THURSDAY, DEC 6:

14.1(a,b,c), 14.2 (b,d,e), 14.5 (a), 14.6, 14.7, 14.12, 14.14