Homework Set #4 Solutions
1. Show that cfw_(x, y) R2 : x, y (0, 1) is open in R2 with the Euclidean metric.
Proof. Let A := cfw_(x, y) R2 : x, y (0, 1). Let (x, y) A be arbitrary. By denition
of A, x (0, 1) and y (0, 1). Since (0, 1) is open in R
Homework Set #6
due Thursday, March 13
1. Suppose f and g are continuous functions and the domain of f is [a, b] and the domain
f (x) if x [a, b]
of g is [b, c]. Show that if f (b) = g(b), then the function h(x) :=
g(x) if x (b, c]
Homework Set #3 Solutions
1. Consider S := R, with the usual metric. Let E :=
: n N . Is E closed, open or
E is not closed. To see this, consider the sequence xn := n . Notice that xn 0, and
0 inE. Thus, there is a sequence in E wh
Homework Set #5 Solutions
1. Suppose that f is continuous at x0 and g is continuous at f (x0 ). We proved in class
that g f was continuous at x0 using sequences. Prove the same theorem using the
criterion for continuity.
Proof. Let > 0 be given.
Homework Set #2 Solutions
1. Suppose (S, d) is a metric space, and let x S and R > 0 both be xed. Show
that BR (x) is open in S. (Hint: draw some pictures for the case of S = R2 with the
Proof. Let z BR (x), and dene r := R d(x
Homework Set #1 Solutions
1. Suppose that (S, d) is a metric space. Suppose E1 , E2 , . . . , Ek are sets that are open in
S. Show that k Ei (the intersection of E1 , E2 , . . . , and Ek ) is also open in S.
Proof. Let x k Ei be arbitrary. Fo
1. Let (S, d) be an arbitrary metric space, and x an arbitrary p S and an arbitrary
R > 0. Show that cfw_x S : d(x, p) > R is open in S.
Proof. Let A := cfw_x S : d(x, p) > R. Suppose q A is arbitrary, and let r :=
d(q, p) R. B
Final Exam Solutions
1. Suppose that
|an an+1 | converges. Show that the sequence an converges.
Proof. Let Sn :=
Notice that since
ak ak+1 be the nth partial sums of
|an an+1 | converges, we know that
absolutely, and therefore