MAT 412 HW 3.3: CONTINUITY
Solution
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 continuo
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MAT 412 HW 3.1: TWO-SIDED LIMITS
Due Friday 02/26
Problem 1. Using the definition of the limit (Definition 3.1) prove
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that
lim (x2 + 5x) = 6
x2
Proof
Given > 0 let = mincfw_1, /2 Since 1 < /
MAT 412 HW 3.2: OTHER NOTIONS OF LIMIT:
ONE-SIDED, INFINITE, AT INFINITY
Solution
Problem 1. Use the definition of limit to find
sin x
lim
x+ x x
if it exists.
Claim: the limit is 0.
Proof: Given > 0
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MAT 412 HW 1.2: PROVING INEQUALITIES
Due Wednesday 01/27 by the end of the day
Problem 1.
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Let a, b R. Which of the following statements are true? Prove the
true ones, and give counterexamples
MAT 412 HW 2.2: LIMIT THEOREMS, SOLUTION
Problem 1. Use the limit theorems to find the limit
(n 1)2
n 3 + 5n2
lim
if it exists.
Solution: Divide numerator and denominator by n2 :
(1 1/n)2
(n 1)2
=
3 +
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
3n 1
=0
n
n2
lim
Proof : Given > 0, let N = d3/ + 1e. If n N , then
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MAT 412 HW 3.4: UNIFORM CONTINUITY
Due Wednesday 03/09.
Problem 1. Give an example of two uniformly continuous functions
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f, g on R such that the product f g is not uniformly continuous on R.
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MAT 412 HW 2.3: MONOTONE SEQUENCES,
BOLZANO-WEIERSTRASS THEOREM
Due Friday 02/19 by the end of the day
Problem 1. Are the following statements true or false? Prove the
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true ones and give cou
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MAT 412 HW 1.3: SUPREMUM AND INFIMUM
Due Monday 02/01 by the end of the day
Problem 1.
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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
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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
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also a Cauchy sequence.
(Hint: you can use Theorem 2.29)
Since cfw_xn