Math 115: midterm 1
May 8, 2013
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justication for everything you are asked to prove.
(10 points)
1. Let F be an or
Math 115: solutions for homework 3
1. a) Rewrite:
n2 + n
1
=
3
3n
3
1
1
+ 2
n n
Example 9.7-(a) says that
lim
n
1
=0
n
and
lim
n
1
=0
n2
Therefore, by theorem 9.3
lim
n
1
1
+ 2
n n
1
1
+ lim 2 = 0 + 0 = 0
n n
n n
= lim
b) Rewrite:
1
1
1
4 + n2 + 2 n + 3
Math 115: solutions for homework 2
1. The hypothesis Ik+1 Ik implies that xk+1 xk , and that yk+1 yk for every
k N. It follows by induction that for k, l N with l k we have xl xk and
yl yk . The non-empty set A = cfw_xl : x N is bounded above by y1 : for
Math 115: homework 2
Due by 2 p.m. on May 2, 2013
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justication for everything you are asked to prove.
1. For each
Math 115: solutions for homework 1
1. a) Repeatedly using the associativity axiom for addition, we get:
A1
(a + (b + c) + (d + e) = (a + b) + c) + (d + e)
A1
= (a + b) + (c + (d + e)
A1
= (a + b) + (c + d) + e)
A3
DL
b) We have a (0 + 0) = a 0 (since 0 +
Math 115: solutions for homework 3
1. a) Fix x0 R. Let = 1. First assume x0 Q, so f (x0 ) = 0. Given any > 0,
/
by density of Q in R we can nd a rational y (x0 , x0 + ) Q. Then
f (y) = 1, so |f (y) f (x0 )| = |1 0| = 1 . We have thus proved f is not
conti
Math 115: homework 4
Due by 2 p.m. on May 30, 2012
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justication for everything you are asked to prove.
1. This co
Math 115: solutions for exam 1
1. Let x F . Then x2 + 2x + 2 = (x + 1)2 + 1. Since y 2 0 for any y F , then
(x + 1)2 0. Therefore,
x2 + 2x + 2 = (x + 1)2 + 1 0 + 1 = 1 > 0
since 1 > 0 (due to 1 = 12 0 and 1 = 0). Therefore, x2 + 2x + 2 = 0 for any
x F.
2.
Math 115: supplemental exercises
In class, we identied the rational numbers with a subset of the real numbers. The
next exercise is meant to show that the rational numbers embed into any ordered
eld.
S1. Let (F, +, 0, , 1, ) be an ordered eld. You are tas
Math 115: homework 3
Due by 2 p.m. on May 16, 2012
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justication for everything you are asked to prove.
1. Find th
Math 115: homework 1
Due by 2 p.m. on April 18, 2013
Notes:
- You may use any result which was stated in class, unless you are asked to prove
it, or explicitly instructed otherwise.
- Give a full justication for everything you are asked to prove.
1. Let (