MAT 412 HW 3.3: CONTINUITY
Problem 1. Suppose that f : R R is a bounded function. (This
means there is a number M such that |f (x)| M for all x R.) Let
g(x) = xf (x). Prove that g is continuous at 0.
Proof Given > 0, let = /(M + 1). If |x| < , th
MAT 412 HW 3.1: TWO-SIDED LIMITS
Due Friday 02/26
Problem 1. Using the definition of the limit (Definition 3.1) prove
lim (x2 + 5x) = 6
Given > 0 let = mincfw_1, /2 Since 1 < /2 we know > 2
Use backslash-brace to make curly braces
MAT 412 HW 3.2: OTHER NOTIONS OF LIMIT:
ONE-SIDED, INFINITE, AT INFINITY
Problem 1. Use the definition of limit to find
x+ x x
if it exists.
Claim: the limit is 0.
Proof: Given > 0, let D = 1/2/3 . If x > D, then
1 < 1 =
MAT 412 HW 1.2: PROVING INEQUALITIES
Due Wednesday 01/27 by the end of the day
Let a, b R. Which of the following statements are true? Prove the
true ones, and give counterexamples to false ones.
(a) If |a| |b|, then |a 1| |b + 1|.
MAT 412 HW 2.2: LIMIT THEOREMS, SOLUTION
Problem 1. Use the limit theorems to find the limit
n 3 + 5n2
if it exists.
Solution: Divide numerator and denominator by n2 :
3 + 5n2
3/n + 5
Since 1/n 0, we have 3/n 0 and 3/n + 5 0 +
MAT 412 HW 2.1: LIMITS OF SEQUENCES
Due Friday 02/05 by the end of the day
Problem 1. Using the definition of the limit, prove that
Proof : Given > 0, let N = d3/ + 1e. If n N , then n > 3/,
hence 3/n < . It follows that
MAT 412 HW 3.4: UNIFORM CONTINUITY
Due Wednesday 03/09.
Problem 1. Give an example of two uniformly continuous functions
f, g on R such that the product f g is not uniformly continuous on R.
(Hint: use an example of a non-uniformly continuous fu
MAT 412 HW 2.3: MONOTONE SEQUENCES,
Due Friday 02/19 by the end of the day
Problem 1. Are the following statements true or false? Prove the
true ones and give counterexamples to the false ones.
(a) If xn is an increas
MAT 412 HW 1.3: SUPREMUM AND INFIMUM
Due Monday 02/01 by the end of the day
Let E = cfw_n/(3n 2) : n N.
(a) Find, with a proof, the supremum of E.
Supremum = 1
Since 1 n, 1 3n 2
So n/(3n 2) n
Therefore since 1 N 1 is an upper bound.
MAT 412 HW 2.4: CAUCHY SEQUENCES
Due Monday 02/22
Problem 1. Suppose cfw_xn is a Cauchy sequence. Prove that cfw_x2n is
also a Cauchy sequence.
(Hint: you can use Theorem 2.29)
Since cfw_xn is a cauchy sequence, it converges to a number a. By