Mathematics Homework Help

2. (a) (4 points) Let xn > 0 for all n 2 N and suppose that lim

n!1

(nxn) = `, where ` 6= 0.

Show that the series

X1

n=1

xn diverges.

(b) (4 points) Let xn > 0 for all n 2 N and suppose that the sequence (n2xn) converges.

Show that

X1

n=1

xn converges.

3. (8 points) Consider the function

f(x) =

(

x 1 if x 2 R is rational;

5 x if x 2 R is irrational:

Show that lim

x!3

f(x) exists but lim

x!a

f(x) does not exist for any a 6= 3.

4. Let f : R ! R and assume there is a constant c 2 (0; 1) such that

jf(x) f(y)j cjx yj

for all x; y 2 R.

(a) (2 points) Show that f is continuous on R.

(b) (4 points) Let x1 2 R and consider the sequence xn+1 = f(xn). Show that (xn)

converges.

Hint: By looking at the dierence jxn+1 xnj, show that the sequence (xn) is

Cauchy.

(c) (2 points) Let lim

n!1

xn = x. Show that x = f(x).

(d) (2 points) Show that x is the unique point such that x = f(x).

5. Let A = f1=n : n 2 Ng.

(a) (4 points) Show that every function f : A ! R is continuous.

(b) (4 points) Is every function f : A ! R uniformly continuous? Either prove this

or give a counterexample (and prove that your counterexample is not uniformly

continuous on A).