Math 554 Sample Homework Assignments

MATH 554 - Math 703 I
Analysis I

Homework Assignments and Supplementary Materials

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Homework Assignments
(Each Problem is worth 10 pts.)

No. Assignment Due Page and No.'s Solutions

#1
Wednesday (1/21)
In problems 1-2, assume F is a field:

Carefully prove that multiplicative inverses are unique.
Carefully prove that (-1)(-1)=1. You can use any propositions which we have proved in class.
In Z₅, what is the (i) additive inverse of 1; (ii) the multiplicative inverse of 3, of 4; (iii) find the solution of the equation 3 x + 1 =2. (Hint: Make addition and multiplication tables and do look-ups.)

#2
Friday (1/30)

Suppose F is an ordered field. If an element a in F satifies |a|< d for each d>0, then carefully prove that a=0.
Page 30 of text: 4(a); 5 (a), (b), (c); 6.; 10 (a), (b)
Extra Credit: 10 (c)

Additional Graduate Student Problems: Page 30: 11

#3
Friday (2/06)

Prove the equivalence statement for "greatest lower bounds" (we proved in class the similar statement for least upper bounds): If L is the greatest lower bound for a set A, then
(i) L is a lower bound for A
(ii) if e>0, then there exists a in A so that a < L+e.
Prove that if L₁ and L₂ are both least upper bounds for a set A, then L₁=L₂. (Moral: least upper bounds are unique)
Using the property that the real numbers satisfy the completeness axiom (i.e. the least upper bound property), prove that R satisfies the property that each nonempty set B, which is bounded from below, must have a greatest lower bound. (Hint: Consider the negatives of sets and the fact that inequalities are reversed when multiplied by a negative.)
Using induction, prove that
1+r+r²+ ... + rⁿ = (1-rⁿ⁺¹)/(1-r)
if r is not equal to 1.
Using the properties of "<", prove that the interval
(a,b) = {x in R | |x-c|< r}
where c=(a+b)/2 and r=(b-a)/2.

#4
Friday (2/13)
Click here to download the assignment.

#5 Monday (2/23) Click here to download the assignment.
Extra Credit (15 pts.)-- Prove the following about closure of sets:

Given a set S, let C be S unioned with all the limit points of S. Show that the complement of C is open. (Hence C is closed.) C is called the "closure" of S.
Suppose that D is a closed set containing S, then prove that the closure of S is contained in D.
Prove that the intersection of all closed sets containing S is the same set as the closure of S.
Main

E.C.

#6 Friday (2/27) Problem: Suppose that S has the property that each sequence from S which converges, must converge to a limit which belongs to S. Prove that S is a closed set.

Extra Credit (10 pts.)-- Complete the proof of the following:
Suppose in the real numbers the sequence {b_n } converges to b and b is not zero. Prove that 1/b_n converges to 1/b.

#7 Friday (3/26) You can download homework assignment #7 for Friday March 26, here.

#8 Friday (4/9) Problem: Using properties of connectedness and the last theorem proved in Monday's lecture (about continuity of inverse functions), prove that for each natural number n:

non-negative n-th roots of non-negative numbers exist and furthermore,
the function g(x) := x^1/n is continuous for x larger or equal zero.

Extra Credit (10 pts.)-- Fill in the details of the proof begun during Monday's lecture:
Prove that a function is continuous if and only if the inverse image of a closed set is closed. Be sure to prove the set equality relating inverse images of the complements of sets.

This page maintained by Robert Sharpley (sharpley@math.sc.edu) and last updated Nov. 2, 2003.
This page ©2003-2004, The Board of Trustees of the University of South Carolina.
URL: http://www.,math.sc.edu/~sharpley/math554

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No.	Assignment Due	Page and No.'s	Solutions
#1	Wednesday (1/21)	In problems 1-2, assume F is a field: Carefully prove that multiplicative inverses are unique. Carefully prove that (-1)(-1)=1. You can use any propositions which we have proved in class. In Z₅, what is the (i) additive inverse of 1; (ii) the multiplicative inverse of 3, of 4; (iii) find the solution of the equation 3 x + 1 =2. (Hint: Make addition and multiplication tables and do look-ups.)
#2	Friday (1/30)	Suppose F is an ordered field. If an element a in F satifies \|a\|< d for each d>0, then carefully prove that a=0. Page 30 of text: 4(a); 5 (a), (b), (c); 6.; 10 (a), (b) Extra Credit: 10 (c) Additional Graduate Student Problems: Page 30: 11
#3	Friday (2/06)	Prove the equivalence statement for "greatest lower bounds" (we proved in class the similar statement for least upper bounds): If L is the greatest lower bound for a set A, then (i) L is a lower bound for A (ii) if e>0, then there exists a in A so that a < L+e. Prove that if L₁ and L₂ are both least upper bounds for a set A, then L₁=L₂. (Moral: least upper bounds are unique) Using the property that the real numbers satisfy the completeness axiom (i.e. the least upper bound property), prove that R satisfies the property that each nonempty set B, which is bounded from below, must have a greatest lower bound. (Hint: Consider the negatives of sets and the fact that inequalities are reversed when multiplied by a negative.) Using induction, prove that 1+r+r²+ ... + rⁿ = (1-rⁿ⁺¹)/(1-r) if r is not equal to 1. Using the properties of "<", prove that the interval (a,b) = {x in R \| \|x-c\|< r} where c=(a+b)/2 and r=(b-a)/2.
#4	Friday (2/13)	Click here to download the assignment.
#5	Monday (2/23)	Click here to download the assignment. Extra Credit (15 pts.)-- Prove the following about closure of sets: Given a set S, let C be S unioned with all the limit points of S. Show that the complement of C is open. (Hence C is closed.) C is called the "closure" of S. Suppose that D is a closed set containing S, then prove that the closure of S is contained in D. Prove that the intersection of all closed sets containing S is the same set as the closure of S.	Main E.C.
#6	Friday (2/27)	Problem: Suppose that S has the property that each sequence from S which converges, must converge to a limit which belongs to S. Prove that S is a closed set. Extra Credit (10 pts.)-- Complete the proof of the following: Suppose in the real numbers the sequence {b_n } converges to b and b is not zero. Prove that 1/b_n converges to 1/b.
#7	Friday (3/26)	You can download homework assignment #7 for Friday March 26, here.
#8	Friday (4/9)	Problem: Using properties of connectedness and the last theorem proved in Monday's lecture (about continuity of inverse functions), prove that for each natural number n: non-negative n-th roots of non-negative numbers exist and furthermore, the function g(x) := x^1/n is continuous for x larger or equal zero. Extra Credit (10 pts.)-- Fill in the details of the proof begun during Monday's lecture: Prove that a function is continuous if and only if the inverse image of a closed set is closed. Be sure to prove the set equality relating inverse images of the complements of sets.



This page maintained by Robert Sharpley (sharpley@math.sc.edu) and last updated Nov. 2, 2003. This page ©2003-2004, The Board of Trustees of the University of South Carolina. URL: http://www.,math.sc.edu/~sharpley/math554