Math 320-2: Homework 2
Northwestern University, Winter 2016
1. Determine whether or not each of the following sequences of functions is pointwise convergent
on the specified domain.
r
xn2 + x
1
n
n
(a) fn (x) = sin(x + 1) 2
on [1, 1] (b) gn (x) = (x 1) 2
Math 320-1: Lecture Notes
Northwestern University, Fall 2013
Written by Santiago Caez
n
These are lecture notes for Math 320-1, the rst quarter of Real Analysis, taught at Northwestern University in the fall of 2013. The book used was the 4th edition of A
Math 320-1: Midterm 2 Practice
Northwestern University, Fall 2014
1. Give an example of each of the following. Justify your answer.
(a) A function on (1, 2) which is continuous but not uniformly continuous.
(b) A function which sends Cauchy sequences to C
Math 320-1: Midterm 2 Practice Solutions
Northwestern University, Fall 2014
1. Give an example of each of the following. Justify your answer.
(a) A function on (1, 2) which is continuous but not uniformly continuous.
(b) A function which sends Cauchy sequ
Math 320-1: Final Exam Practice Solutions
Northwestern University, Fall 2014
1. Give an example of each of the following. Justify your answer.
(a) A nonconstant and nonnegative integrable function on [0, 1] with integral zero.
(b) A nonintegrable function
Math 320-1: Homework 1
Northwestern University, Fall 2014
1. Wade, 1.2.7bd. Let x R.
(b) Prove that |x| 1 implies |x2 + 2x 3| 4|x 1|.
(d) Prove that 1 < x < 0 implies |x3 2x + 1| < 1.26|x 1|.
2. Wade, 1.3.3. Prove that if a < b are real numbers, then ther
Math 320-1: Final Exam Practice
Northwestern University, Fall 2014
1. Give an example of each of the following. Justify your answer.
(a) A nonconstant and nonnegative integrable function on [0, 1] with integral zero.
(b) A nonintegrable function f on [3,
Math 320-3: Lecture Notes
Northwestern University, Spring 2015
Written by Santiago Ca
nez
These are lecture notes for Math 320-3, the third quarter of Real Analysis, taught at Northwestern University in the spring of 2015. The book used was the 4th editio
Math 320-2: Homework 4 Solutions
Northwestern University, Winter 2016
1. Recall that the taxicab metric d1 on R2 is defined by
d1 (x1 , y1 ), (x2 , y2 ) := |x1 x2 | + |y1 y2 |
and the box metric d2 is defined by
d2 (x1 , y1 ), (x2 , y2 ) := maxcfw_|x1 x2
Math 320-2: Homework 3
Northwestern University, Winter 2016
1. Wade, 7.1.7. Suppose that f is uniformly continuous on R. If yn 0 as n and
fn (x) := f (x + yn ) or x R. Prove that fn converges uniformly on R.
2. Wade, 7.2.5. Show that
X
1
x
f (x) =
sin
k
k
Math 320-2: Homework 6 Solutions
Northwestern University, Winter 2016
1. Show that any compact metric space K has a countable dense subset. Hint: For each n N,
the collection cfw_B1/n (p) of all possible open balls of radius n1 as p varies throughout K is
Math 320-2: Homework 3 Solutions
Northwestern University, Winter 2016
1. Wade, 7.1.7. Suppose that f is uniformly continuous on R. If yn 0 as n and
fn (x) := f (x + yn ) for x R. Prove that fn converges uniformly on R.
Proof. For a fixed x R, since yn 0 a
Math 320-2: Homework 1 Solutions
Northwestern University, Winter 2016
P
1. Suppose
that an 0 for all n and that (bn ) is a bounded sequence. If
an converges, show
P
that
an bn converges. Also, give an example showing this is not necessarily true without t
Math 320-2: Homework 1
Northwestern University, Winter 2016
P
1. Suppose
that an 0 for all n and that (bn ) is a bounded sequence. If
an converges, show
P
that
an bn converges. Also, give an example showing this is not necessarily true without the
assumpt
Math 320-2: Homework 5 Solutions
Northwestern University, Winter 2016
1. Suppose that U and V are open subsets of R. Show that U V is open in R2 . Here we are
considering R with the standard metric, but feel free to use whichever of the Euclidean, taxicab
Math 320-1: Homework 2
Northwestern University, Fall 2014
1. Wade, 2.1.0bcd. Decide which of the following statements are true and which are false. Prove
the true ones and provide a counterexample for the false ones.
(b) If xn does not converge, then xn /