Homework due on Monday, Sept. 14
2.1: 1ab, 2ab, 3cf, 5bc, 11, 19ab
2.2: 3afikn, plus use dual inclusion to prove parts l and n of Theorem 2.6, and part g of Theorem 2.7
2.2: 14abef,15ab (Note: Regarding problem 14, providing a "counterexample" means giving a specific choice of sets for which the assertion is false.) Also, if A and B are sets, use "set algebra" to simplify A-(A-B). Be sure to justify your steps by quoting a rule that applies.
Note: You may use "proof by contradiction" if you wish. The principle involved in this method of proof is that the statement "the truth of P implies the truth of Q" is logically equivalent to the statement "if Q is false then P is false".
Proof by contradiction should be used for the following: 1.5:3dg,5 plus show that sqrt{3} is not a rational number; here, "sqrt" stands for "square root".
Homework due Monday, Sept. 21
1.3: 1j,2j,6ce
Plus:
Use the definition to prove that lim_{n \to \infty} (3n-10)/n = 3; and prove that lim_{n \to \infty} (3n-10)/n is not 7/2 by using the negation of the definition.
Give the negation of lim_{n \to \infty} a_n = + \infty by using a properly quantified English sentence.
Use the definition to prove that lim_{n \to \infty} sqrt{n^2-2}= + \infty, and that lim_{n \to \infty} (n^2 -100n)^{1/3} = +\infty.
OUR FIRST EXAM WILL BE HELD ON WEDNESDAY, OCT. 14, DURING THE LECTURE PERIOD
Homework due Monday, Sept. 28
(1) State and prove an improved version of the Squeeze Theorem for the case when the limits are -\infty. The version that I stated in class is correct, but some of its hypotheses are not needed.
(2) Show that it is not generally true that lim_{n \to \infty}a_n = L if lim_{n \to \infty} |a_n| = |L|. Assume that L is finite.
(3) The triangle inequality states that if x and y are real numbers, then |x+y| \le |x|+|y|. Here, the symbol "\le" stands for "less than or equal to". Use the triangle inequality to prove that if x and y are real numbers, then both |x| -|y| \le |x-y| and |y|-|x| \le |x-y| are true, or equivalently, that | |x|-|y| | \le |x-y|.
(4) Use the definition to prove that lim_{n \to \infty} (sqrt{n+1} - sqrt{n}) =0.
(5) Give examples of sequences {a_n} and {b_n} with limits +\infty such that the sequence {a_n-b_n} has limit 1, has limit +\infty, and has limit -\infty. Of course, there should be different examples for each conclusion. Justify your examples by giving proofs based on the definitions.
(6) Prove the first two parts of the Simple Limit Theorem, namely
(a) If {a_n} is a sequence of real numbers, c and L are real numbers, and lim_{n \to \infty} a_n = L, then lim_{n \to \infty} ca_n = cL. (Hint: Consider the case c=0 separately.)
(b) If {a_n}, {b_n} are sequences of real numbers with finite limits, then lim_{n \to \infty} (a_n+b_n) = lim_{n \to \infty} a_n + lim_{n \to infty} b_n.
Homework due Monday, Oct. 5:
(1) Prove that the sequence {(-1)^n} does not converge.
(2) If {a_n} is a sequence of real numbers with lim_{n \to \infty} a_n >0, show that there exist \epsilon >0 and an integer N so that a_n > \epsilon if n >N. (This is a slight variant of a lemma we proved in class.)
(3) Suppose that {a_n} and {b_n} are sequences of real numbers satisfying lim_{n \to \infty} a_n >0 and lim_{n \to \infty} b_n = +\infty. Prove that lim_{n \to \infty} a_nb_n = +\infty. (Use the definition; this is not a corollary of the Simple Limit Theorems.)
2.4: 8ejlt
Homework due Monday, Oct. 12:
2.4: 9df
Plus: (1) For any two nonnegative integers n and k with k \le n, define the binomial coefficient b(n,k) = n!/[k!(n-k)!]. Here we adopt the standard convention that 0! =1. Use arithmetic to prove that if n and k are integers with 1 \le k \le n, then b(n+1,k)= b(n,k) + b(n,k-1).
(2) Use induction to prove that for any two real numbers x,y and all positive integers n, it is true that (x+y)^n = \sum_{k=0}^n b(n,k) x^{n-k}y^k. This formula is a special case of what is called the binomial expansion.
(3) Use case analysis to show that for any two real numbers x and y, it is true that |xy| = |x||y|.
Homework due Monday, Oct. 19:
None!
Homework due Monday, Oct. 26:
3.1: 3df, 4a, plus: If A X B = A X C and A is not the empty set, show that B=C. Be sure that you indicate how you use the fact that A isn't empty.
3.2: 1dhi, 2bc, 4aeh
Homework due Monday, Nov. 2:
3.3: 2abc, 3bc, 6bc, 7
1.7: 8b, 9abc, 10
Homework due Monday, Nov. 9:
6.1: 13
Plus: (1) Make Cayley tables for Z_8 for both mutiplication mod 8 and addition mod 8. Identify all elements of Z_8 which have inverses with respect to the operation of multiplication mod 8.
(2) For n in N, let S_n be the collection of all permutations of n letters. Use induction to show that S_n has exactly n! distinct elements.
(3) If (G,#) is a group and if a#a = e (the identity) for all a in G, show that (G,#) is Abelian.
6.2: 14,16
OUR SECOND EXAM WILL BE ON WEDNESDAY, NOV. 18, DURING THE REGULAR LECTURE TIME.
Homework due Monday, Nov. 16:
6.5: 2
Homework due Monday, Nov. 23:
(1) Find the inverse of the complex number 2+3i. Find the inverse of a general complex number a+ib which is not zero.
(2) Find the real and imaginary parts of (1-i)/(2+3i).
(3) Prove DeMoivre's formula: If n is a natural number and x is a real number, then (cos x +i sin x)^n = cos nx +i sin nx.
Homework due Monday, Nov. 30:
(1) Verify by using the definition of limit that lim_{x \to -1} 1/[x(2x+1)] = 1.
(2) Verify by using the definition that lim_{x \to 0-} 1/[x(2x+1)] = -\infty.
(3) Verify by using the definition that lim{x \to +\infty} 1/[x(2x+1)] = 0.
(4) Let x_0 be a real number. What is the logical negation of lim_{x \to x_0} f(x) = +\infty?