# Assignments

### Assignment 6

NOT TO BE HANDED IN
Solutions will be handed out on Wednesday March 29.
• Section 8.1 numbers 2, 4, 12, 14, 18, 22, 23
• Section 8.2 numbers 1, 2, 5
• Section 9.4 numbers 6c, 8, 19
• Let be a doubly indexed sequence. Consider the question of whether or not .
• Show that if is bounded and increasing (that is, and imply that ) then the two limits are equal.
• Show that the two limits may not be equal otherwise. [Hint, consider ].

### Assignment 5

Due Monday March 20, 4:30 PM
• Section 6.2 numbers 8, 10, 19, 20
• Section 6.4 number 16
• Read page 222-226 on Newton's method, and do the following:
• Assume that f' doesn't vanish. Show that r is a root of f if and only if it is a fixed point of the function g(x) = x - f(x)/f'(x).
• Apply Taylor's theorem to g in order to show (**). That is, show that if and if , then there is a K such that , for x near r.
[Note that this implies that convergence to r is very fast.]
• Apply the mean value theorem to g to show that it is a contraction on some interval about r. [Note that, this gives another way of showing convergence, based on results earlier in the course.]

### Assignment 4

Due Friday March 3, 4:30 PM
• Section 10.3 numbers 1,7,8
• Section 5.2 number 12
• Section 5.4 numbers 3,4,6,8

### Assignment 3

Due Wednesday February 23, 4:30 PM
Postponed to Friday February 25
(A prettier looking version of this assignment was handed out in class)

Definition: Let U be a subset of d-dimensional Euclidean space.

• An "interior point" of U is any point u such that B(u,r) is contained in U, for some r>0. The set of interior points of U is called the "interior" of U.
• A "limit point" of U is any point that is the limit of a sequence of points of U. The set of limit points of U is called the "closure" of U.
Show the following properties. You may do so in whatever order you find most convenient, and once you have proven one fact, you may use it in the proofs of other facts. Be on the lookout for statements that follow from other statements with little extra work.
• The interior of the complement of U equals the complement of the closure of U.
• The closure of the complement of U equals the complement of the interior of U.
• The interior of U is an open subset of U.
• If V is any open subset of U then V is contained in the interior of U.
• U is open if and only if it equals its interior.
• The closure of U is a closed set containing U.
• If V is any closed set containing U then V contains the closure of U.
• U is closed if and only if it equals its closure.
• The interior of U is disjoint from the boundary of U.
• The closure of U is the union of the interior of U and the boundary of U.
• The interior of the interior of U equals the interior of U.
• The closure of the closure of U equals the closure of U.
• The interior of the intersection of U and V equals the intersection of their interiors.
• The interior of the union of U and V contains the union of their interiors.
• Give an example where this containment is not an equality.
• The closure of the union of U and V equals the union of their closures.
• The closure of the intersection of U and V is contained in the intersection of their closures.
• Give an example where the latter contaiment is not an equality.
• Page 354, problems 6 and 7.

### Assignment 2

Due Monday Jan 31, in class (I will give out solutions in class)
• Section 3.4, numbers 1, 5, 7, 9
• Show that lim x_n = x if and only if limsup x_n = x = liminf x_n
• Show that limsup -x_n = -liminf x_n

### Assignment 1

Due Friday Jan 21, by 4:30 PM
• Section 3.3, number 7
• Section 3.5 numbers 2a, 3a, 4
• Let f map [0,1] to [0,1], and be continuous and increasing. Suppose it has a unique fixed point b. Show that, whatever initial value x_1 is chosen (from [0,1]), the iteration x_{n+1}=f(x_n) gives a sequence that converges to b.